Where has this common generalization of nets and filters been written down? It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters.  They are of course essentially equivalent, but each has its own minor advantages for pedagogy and intuition.  Seemingly less well-known is the following common generalization of both nets and filters:

Let $X$ be a topological space.  Then a filternet (this is a term I made up) in $X$ consists of a set $I$, a filter $F$ on $I$, and a map $I\to X$ (written $i\mapsto x_i$).  We say $(x_i)$ converges to a point $x\in X$ if for each neighborhood $U$ of $x$, $\{i\in I:x_i\in U\}\in F$.

If $I$ is a directed set, then we can take $F$ to be the "eventual" filter on $I$, and then $(x_i)$ converges to $x$ as a filternet iff it converges to $x$ as a net.  On the other hand, if $F$ is a filter on $X$, we can take $I=X$ and $x_i=i$ for all $i$, and $(x_i)$ converges to $x$ iff the filter $F$ converges to $x$.  So filternets include both filters and nets as special cases.  Like filters and nets, filternets can be used to describe basic topological notions (closed sets, continuity, compactness, etc.) in terms of convergence.  Pedagogically, filternets have some of the advantages of both filters and nets: like nets, they are intuitively similar to sequences (they have an index set, and it is obvious how to push them forward along maps), and like with filters, the theory of "subfilternets" and "ultrafilternets" is very simple and does not require you to change the index set.
While nets and filters are both quite well-represented in the literature, filternets are not so common.  The main way I have seen them used is in talking about limits of sequences with respect to an ultrafilter on $\mathbb{N}$, usually in the context of talking about ultraproducts of metric spaces and related ideas (see this Wikipedia page, for instance).  But this is still rather different from thinking of them as providing a general theory of convergence like nets and filters, and in fact I have never seen filternets used in this general way in any published work (I learned about them as a general theory of convergence from Nik Weaver, who just called them "nets", in my undergraduate pointset topology class).  So, my question is:

Where have "filternets" been written about (as a general theory of convergence on topological spaces)?  Is there a standard name for them?  Who first invented them?  Is there some kind of standard reference that covers them (e.g., a pointset topology textbook that uses them, or some well-known expository paper that discusses them alongside filters and nets)?

 A: I can't say who was first, but I can say who was (at this moment, at least) last: me.  Less than 24 hours ago, Misha Kapovich sent me a modified version of his striking result that for a connected $\mathbb{C}$-manifold $M$, the ring $\operatorname{Hol}(M)$ of holomorphic functions on $M$ either consists of the constant functions or has infinite Krull dimension.  His idea to repair a faulty argument of someone else's was to use ultralimits.  This was something that I had probably seen before but had to re-look up the definition.  Moreover, since I was writing up a proof of Kapovich's Theorem for (the next iteration of) my commutative algebra notes, I had to say something about ultralimits.  Here is what I say to introduce them, copied and pasted directly:
$\newcommand{\ra}{\rightarrow} \newcommand{\N}{\mathbb{N}}$

Let $I$ be a set, let $X$ be a topological space, and let $x_{\bullet}: I \ra X$ be an $I$-indexed sequence -- i.e., a function!  Let $\mathcal{F}$ be an ultrafilter on $I$.  We say $x \in X$
  is an ultralimit of $x_{\bullet}$ and write $\mathcal{F}\lim x_{\bullet} = x$ if $x_{\bullet}(\mathcal{F}) \ra x$: that is, for every neighborhood $U$ of $x \in X$, we have $x_{\bullet}^{-1}(U) \in \mathcal{F}$.  From the general theory of filter convergence, we deduce: (i) If $X$ is Hausdorff, then every $I$-indexed sequence $x_{\bullet}: I \ra X$ has at most one
  ultralimit. (ii) If $X$ is quasi-compact, then every $I$-indexed sequence has at least one ultralimit.  Thus (iii) If $X$ is compact, then every $I$-indexed sequence has a unique ultralimit.
  In our application we will have $I = \N$, $\omega$ a fixed nonprincipal ultrafilter and $X = [0,\infty]$. Thus we have an ordinary sequence $\{x_n\}$ in $[0,\infty]$, and $\omega \lim x_n = x$ means: for all $\epsilon > 0$, the set of $n \in \N$ such that $|x_n-x| < \epsilon$ lies in $\omega$.  Because $[0,\infty]$ is compact, any sequence in $[0,\infty]$ has a unique ultralimit.

So there you have "ultrafilternets"...written down on the fly.  OK, some comments:
(i) In the application that comes next, we "fix" a nonprincipal ultrafilter on $\mathbb{Z}^+$, and the fact that $\omega$ must contain the Frechet filter -- i.e., all cofinite sets -- is of course important.  The fact that a "Frechet filter limit" is just a usual sequential limit must be a standard observation among those who work with ultralimits.  I really like the observation that in any directed set $(I,\leq)$ the "principal upsets" $U(i_0) = \{i \in I \mid i \geq i_0\}$ form the base for a filter which plays the analogue of the Frechet filter and that pushing forward this "$I$-Frechet filter" gives you $I$-convergence!
(ii) I think there may be some precedence for your filternets coming from the fact that filters can be defined with respect to any partially ordered set $(I,\leq)$, namely a nonempty subset $\mathcal{F}$ of $I$ which is upward-closed and downward directed: see wikipedia for details.  A principal filter is just an upset $U(i_0)$ for some $i_0 \in I$.  Moreover $I$ is directed iff the family $\{U(i_0) \mid i_0 \in I\}$ is a filter base on $2^I$, in which case it is the base for the $I$-Frechet filter described above.  Hmm...I didn't reach a dramatic conclusion, but it seems like there might be something going on here.
(iii) I think one reason that your filternets are not more commonly known is a familiar, stupid one: most standard general topology texts discuss filters or nets; some give annoyingly perfunctory exercises on the one they didn't cover.  I don't know any standard text that makes concerted use of filters and nets together.   
And finally, two questions.
(A) Other than generalizing filters, nets and ultralimits, do you have a nice application of your filternets?
(B) Are there also netfilters?  (I'm not sure whether I'm serious.  But your construction is a way to build nets faithfully inside the filter framework along with other things besides.  Or maybe netfilters are just nets: to a net one can associate its filter base of tails, and to a filter base one can associate a certain net; going from filter bases to nets back to filter bases recovers the filter base we started with, but going from nets to filters back to nets does not and could not, for cardinality reasons.  Hmm...)
A: Per the OP's request from this comment I am posting some stuff about references I am aware of. Those references are mentioned in my (unfinished) notes on this topic and also in this answer and this conversation in chat. The purpose of reposting them here is that the linked resources have somewhat different scope and focus. Another reason is that if I list there references here, they will become better organized.
I repeat here some stuff mentioned already in Brian M. Scott's answer. But maybe it is not that bad to have one brief and concise answer and also a longer post which digresses a bit to some related stuff.
Since I mention results from various sources, terminology is a bit mixed in this post. But I always use the name filternet in the way you defined it. (This helps to unify terminology a bit; I mention some other places where this notion or some special case is defined, but several different names are used for the same concept in the literature.) If I mention other concepts, I hope it will be clear from the context which type of convergence of filters I mean.
General approach
As already mentioned in Brian M. Scott's answer, Bourbakists used the notion of filternet (in your terminology) in some of their works as a basic notion, which was defined first. (To be more precise, they use filter base more often than filter, but this difference is a very minor one.)
They also define other notions related to convergence (such as cluster point or limit superior) in this generality and various types of limits  were defined as special cases.
You could probably find this in several of their books. I am aware of this approach being used in:


*

*N. Bourbaki. Elements of Mathematics. General Topology. Chapters I-IV. Springer-Verlag, Berlin, 1989. See p.70 for the definition of limit of a filternet.

*Jacques Dixmier. General Topology. Springer-Verlag, New York, 1984. Undergraduate Texts in Mathematics.


In Dixmier's book this approach is very prominent. Already in Chapter II the notion of limit along a filter base is defined. (This is precisely your filternet.) Where possible, other notions are defined in a similar manner and results about them are shown in this generality.
Cluster points (which the author calls adherence values) are defined in the same chapter. Limit superior and limit inferior is defined in Chapter VII (for $X=\mathbb R$). So if you are looking for a book which really uses the notion of filternet to build all theory related to convergence, this is the book I would certainly recommend.1
Other references which mention this type of convergence are:


*

*D. H. Fremlin. Measure theory, Volume 2: Broad Foundations. Torres Fremlin, Essex, 2001. In this book there is one paragraph 2A3S in the appendix, where this notion is defined. Some basic results about filternet are shown here. The corresponding notion of limit superior is defined here, too.


There are also some textbooks in mathematical analysis, which introduce the filternet:


*

*Vladimir A. Zorich. Mathematical Analysis I, Springer, 2004, Undergraduate Text in Mathematics. In this book only the case $X=\mathbb R$ is discussed; see Definition 3.2.12. Again, the idea is to use this as a unifying notion for several other special cases.

*The book M. Gera, V. Ďurikovič: Matematická analýza (Mathematical Analysis, in Slovak) also introduces this notion in the case $I\subseteq\mathbb R$ and $X$ Hausdorff. 


Both these books use the notion of filterbase (similarly as Bourbaki). I am aware that the latter will probably not be useful for you (unless you speak Slovak). But I am posting these two references mainly as evidence that this approach in the introductory analysis textbooks is probably not that uncommon and it is likely that some other similar texts exist. (Although it is probably not easy to find them. I knew about one of them since it was the book from  which I was taught my second course in analysis. And I found the other one using Google Books.) The reason that this approach appears in some texts is, in my opinion, influence of Bourbaki which lead to striving to greatest possible generality. But I have admit that some proofs might be more economical in this setting.
Gera and Ďurikovič do not have any book by Bourbaki among references. In Zorich I have seen some books by Bourbaki, Cartan and Dieudonné. Maybe it would be possible to check the reference given in these books to see from where the authors took this approach. But this would be rather time-consuming undertaking.
Reduction of the general approach to limit of a filter.
Let me also briefly mention the comparison of the two notions:


*

*Your definition of filternet.

*Convergence of filter $\mathcal F$ on a topological space $X$ defined as: The filter $\mathcal F$ converges to a points $x$ iff it contains all neighborhoods of the point $x$.


The latter is the notion which, as far as I can say, appears quite frequently in texts on general topology. And it is rather clear that it is a special case of the former (as was explained already in your question.)
But it might be perhaps useful to notice that also the notion of filternet can be obtained relatively easily using the notion of limit of a filter. Namely if we have filternet $(\mathcal F,f\colon I\to X)$, then this filternet converges to the point $x$ if and only the filter generated by $\{f[F]; F\in\mathcal F\}$ converges to $x$.


*

*This is also discussed in comments to this question.

*This is shown in my notes in the section named "$\mathcal F$-limits and convergence of filters". It is also mentioned in the answer which I have linked to several times.

*This is the way the filternet is defined in Fremlin's book.



The books mentioned above are the only references I am aware of which work in the same generality as suggested in your post. Let me allow to digress a bit from the question to mention two special cases, which are relatively important (and for which there is much more literature).
Ultrafilters and Stone-Čech compactification
In the case that $\mathcal F$ is an ultrafilter on $i$ and $X$ is a compact Hausdorff space, then the convergence  of a filternet $\mathcal f: I\to X$ can be also interpreted in this way: If we endow $I$ with the discrete topology, then we can use the Stone-Čech compactification $\beta I$ on this space. The points of $\beta I$ correspond to ultrafilters on $I$. And the limit of an filternet $(\mathcal F, f\colon I\to X)$ is precisely the value of the unique extension of $f$ to $\overline f \colon \beta I \to X$ evaluated at $\mathcal F$. This is explained in more detail in my other answer.
If we know about this connection, it should not be too surprising that the notion of filternet also appears in some texts studying the Stone-Čech compactification. I will mention the two books where I have encountered this:


*

*N. Hindman, D. Strauss: Algebra in the Stone-Čech compactification; first edition (1998), second edition (2012).

*Stevo Todorčević. Topics in Topology. Springer-Verlag, Berlin{Heidelberg, 1997. Lecture Notes in Mathematics 1652.


In Todorčević's book the notion of filternet is defined in the same way as in your post. (Definition 2 in Section 14.) However, both this notion and the Stone-Čech compactification are used only in one chapter of this book. Although this book only works with $I=\mathbb N$, it is mentioned several times in this chapter that everything works the same for arbitrary set.
The book of Hindman and Strauss is devoted solely to the Stone-Čech compactification. Therefore it devotes a bit more space to filternet than Todorcevic's book. Again, the definition given in this book is exactly the same as yours - see Definition 3.44.
Sequences
If we take $I=\mathbb N$, then we are dealing with sequences. For this case you can find more references. 
The case when $\mathcal F$ is ultrafilter and the sequence is bounded is often mentioned in set-theoretical texts. This Google Books search returns Hrbacek-Jech and Komjáth-Totik, to mention just two examples. 
The first book where I have seen this notion we Balcar-Štěpánek: Teorie Množin (Set Theory, In Czech). I guess that there are some other texts from this area which mention this special case, i.e. limits of bounded sequences along ultrafilters.

Names for this notion
Let me also list here various name which are used for filternet in literature. (To some extent as a curiosity, but perhaps this might be useful when searching for other resources on the topic. Or maybe it might show that searching for something like this on internet might be rather difficult, because every author uses different terminology.)


*

*Bourbaki calls it limit of $f$ with respect to a filter.

*Dixmier uses the name limit along a filter base.

*Zorich chose limit of a function over a base.

*Hindman and Strauss use the name $p$-limit. Similarly Komjáth and Totik use $D$-limit, Hrbacek and Jech $U$-limit. Of course, in each of these there case $p$, $D$ and $U$ denotes an ultrafilter

*Todorčević does not use any name for this notion, he only introduces the notation $\lim\limits_{n\to\mathcal U} x_n$. Fremlin does not have a name for this notion either and he uses the notation $\lim\limits_{x\to\mathcal F} \phi(x)$.



1 Dixmier does not define the notion of Cauchy function with respect to filter base. A possible approach to Cauchyness is briefly explained in my notes. This is the only notion which appears in these notes for which I do not have a reference in a textbook. A special case of Cauchy condition for $I=X=\mathbb R$ is mentioned in the book by Gera and Ďurikovič. 
A: The fullest treatment that I’ve seen is in some notes by our own Martin Sleziak; his site is here, and the notes in question are here [PDF]. He simply calls it $\mathscr{F}$-convergence and says that the only books known to him that develop convergence using it are:

*

*N. Bourbaki. Elements of Mathematics. General Topology. Chapters I-IV. Springer-Verlag, Berlin, 1989.

*Jacques Dixmier. General Topology. Springer-Verlag, New York, 1984. Undergraduate Texts in Mathematics.

I have the Bourbaki. The relevant definition is Definition $3$ in Section $I.7.3$:

Let $f$ be a mapping of a set $X$ into a topological space $Y$, and let $\mathfrak{F}$ be a filter on $X$. A point $y\in Y$ is said to be a limit point (or simply a limit) (resp. cluster point) of $f$ with respect to the filter $\mathfrak{F}$ if $y$ is a limit point (resp. cluster point) of the filter base $f(\mathfrak{F})$.
Th relation “$y$ is a limit of $f$ with respect to the filter $\mathfrak{F}$” is written $\lim_{\mathfrak{F}}f=y$, or $\lim\limits_{x,\mathfrak{F}}f(x)=y$, or $\lim\limits_xf(x)=y$ if there is no risk of confusion.

Added: Martin notes that more information and references can be found in this answer and this conversation in chat.
A: 

Let $X$ be a topological space. Then a filternet (this is a term I made up) in $X$ consists of a set $I$, a filter $F$ on $I$,
and a map $I\to X$ (written $i\mapsto x_i$). We say $(x_i)$ converges to a point $x\in X$
if for each neighborhood $U$ of $x$, $\{i\in I:x_i\in U\}\in F$.

Isn't a filternet merely a continuous map $I\sqcup_{F} \{\infty\}\to X$ where
the topology on $I\sqcup_{F} \{\infty\}$ is such that $F$ is the filter of neighbourhoods of $\infty$,
and $\infty$ is a closed point, and the induced topology on $I$ is discrete?
Explicitly, a subset $U$ of $I\sqcup_{F} \{\infty\}$ is open iff
it either $\infty\not\in U$ or $U\setminus\{\infty\}\in F$.
By the very definition of continuity, the map $(x_i),x_\infty=x: I\sqcup_{F} \{\infty\}\to X$ is continuous iff for each neighbourhood $U$ of $x$, $\{i\in I:x_i\in U\}\in F$, i.e. $(x_i)$ converges to a point $x\in X$.
This space is defined in Bourbaki, General Topology, \S6.5, Def.5, Example, see the picture above. In \S7.5 they do discuss convergence in terms of the trace filters but do not mention explicitly this construction. And I do not think I havev seen a reference explicitly discussing convergence in terms of these maps.
