What's an overneighbourhood? An overneighbourhood of a point $x$ is defined as an overset of a neighbourhood of $x$. What's an overset?
Are these terms out of date? I can't find any reference to them online. (I'm skimming over Ash's "Real analysis and probability", 1972)
 A: Overset
An overset is a superset. That much is mentioned in Ash's book (page $\text{xi}$). I have seen the notion of an overset being used in some textbooks that were translated from Russian : but I can't recall any of them, nor say with confidence that such a term is the result of a Russian translation. However, I can be confident that this IS used somewhere and is still prevalent.

Overneighbourhood
The definition of an overneighbourhood in the sense of Ash may be found on page number $\text{xiv}$.
The reason why the term "overneighborhood" is used is because the definition of neighborhood in Ash's book is different from other books, and to compensate for that, he defines an overneighborhood.
In the book "Elementary Topology Problem Textbook" by Viro, Ivanov, Kharlamov and Netsvetaev, the authors write :

A neighborhood of a point is any open set containing this point. Analysts
and French mathematicians (following N. Bourbaki) prefer a wider notion
of neighborhood: they use this word for any set containing a neighborhood
in the above sense.

It seems that Robert Ash , the book above, and a book like "Real and Functional Analysis" of Bogachev and Smolyanov , all use the definition that a neighbourhood of a point is an open set containing that point. So for example, $[-1,1]$ cannot be a neighbourhood of $0$ under this definition.
However, the two books other than that of Robert Ash also do not require sets with the property that "they contain neighbourhoods" : in every such instance, they are able to , without loss of generality, work with the open set itself. Hence, they see no reason to define an "overneighbourhood".
On the other hand, Ash clearly extends results involving open sets, to overneighbourhoods. For example, look at Theorem 3.5.1 from the book : under four conditions, one can create a "base of overneighbourhoods" for a topological vector space. Or consider the exercise 19(b), Chapter 3 on bornological topological vector spaces : one does not insist that a bornivore be an open set, hence a "neighbourhood" of $0$ in the traditional sense.
To see more such subtleties, this post looks a good bet.
Note that a "neighborhood" of a point may also be defined as just open balls around that point of non-zero radius as well. This is the case in Rudin's Principles of Mathematical Analysis. However, he has no use of overneighbourhoods , because he finds the use of such sets to be unnecessary in that book.
However, in more recent literature, neighbourhoods of a point are defined to be sets that contain some open set around that point. As a consequence, the role of overneighbourhoods is obsolete.
For example, consider the book of Bartle and Sherbert that defines a neighbourhood of a point to be a set that contains an open ball around the point. Specifically, on page $312$,

A neighborhood of a point $x\in \mathbb R$ is any set $V$ that contains an $\epsilon$-neighborhood[sic] $V_{\epsilon}(x) := (x-\epsilon,x+\epsilon)$ of $x$ for some $\epsilon>0$.
While an $\epsilon$-neighborhood is required to be "symmetric about the point", the idea of a (general) neighborhood relaxes this particular feature, but often serves the same purpose.

Note that Willard/Bourbaki's "General Topology" uses the "relaxed" version of neighborhood, while Munkres' Topology uses the same version as Ash and Bogachev-Smolyanov.
However, to all intents and purposes, the term "overneighbourhood" can safely be replaced by the definition of neighbourhood as in the book of Willard/Bourbaki. It wouldn't change a thing, and I assume that you can work with that relaxed definition of neighbourhood safely in the context of Ash's book.
Remark : Thank you @Ulli for pointing out that another popular reference, the book of Ryszard Engelking titled "General Topology", requires neighbourhoods to be open sets (unlike Bartle-Sherbert or Bourbaki).
Since it is a fairly well-cited book, I'll cite exactly what is said in
this book. (Page 12, Section 1.1)

If for some $x \in X$ and an open set $U \subset X$ we have $x \in U$, we say that $U$ is a neighbourhood of $x$.

That is, Engelking also assumes that neighbourhoods are open. Given that this is a popular and well-cited book, I could not afford to leave it out. Note that Engelking avoids the "relaxed" usage of neighbourhoods by never needing to refer to these sets : in a style similar to Ash.
However, I am pretty sure that we won't be hearing of overneighbourhoods ever again : and thank heavens, for that is quite an ungainly name.
