Does the naive definition of "commutative category" have any interesting consequences? By a commutative monoid, let us mean a monoid $A$ in which $a,b \in A$ implies $ab=ba$. Its not at all obvious how to generalize this to the case of an arbitrary category; we cannot just assume that all morphisms $f$ and $g$ satisfy $fg=gf$, since the domain/codomain conditions won't be met. We can however assume that for all objects $X$ and all endomorphisms $f,g \in \mathrm{End}(X),$ $fg=gf.$ Lets call this the naive definition of "commutative category."
Does it have any interesting consequences?
 A: I don't think that this is a useful definition. You don't consider the morphisms which are no endomorphisms.
Commutative monoids are precisely the monoid objects in the category of monoids (Eckmann-Hilton argument). If $M$ is a commutative monoid, the multiplication map is a homomorphism $M \times M \to M$.
Since categories are many-objects-monoids, we could expect that (small) "commutative categories" are the monoid objects in the category of (small) categories, i.e. the strict monoidal categories. This has the correct decategorification: A monoidal category with one object (even not assumed to be symmetric) is the same as a commutative monoid.
I have to admit that this isn't a satisfactory answer, because we would rather expect that "commutative" is a property, not an extra structure.
Edit: There is a good reason why there is no natural definition of a commutative category: If $O$ is a set, the category of $O$-graphs has as objects pairs $(M,M \rightrightarrows O)$, consisting of a set $M$ and a pair of parallel morphisms $s,t : M \to O$. This category is monoidal with unit $(O,O = O)$ and tensor product $(M,s,t) \otimes (M',s',t') = (M \times_{t,s'} M', s \circ \mathrm{pr}_1,t \circ \mathrm{pr}_2)$. Then a category with object set $U$ is the same as a monoid object in this monoidal category (this is easy to see, for a reference see II.7 in Mac Lane's CWM). In order to define commutative monoids, we need a symmetric monoidal category. However, the category of $O$-graphs is not symmetric monoidal. Only the full subcategory consisting of those $O$-graphs with $s=t$ is symmetric monoidal. The commutative monoid objects therein correspond to those categories which are disjoint unions of commutative monoids.
A: I have arrived here from Google in the search of a reference on such categories, so would be very happy if someone turns up some "interesting consequences".
The property that such a "commutative category" might have that I want to take advantage of is that you can actually compute them from a finite presentation.  In fact I require a slightly stronger condition, that the endomorphism monoids are abelian groups.  When I say 'presentation' I have in mind a simplicial set, and I am interested in the fundamental category.  Now categories are too hard to work with for the application I have in mind and I don't want to invert any more edges than I have to.  Imposing the commutativity and invertibility relations on endomorphisms gives a category which is much richer than the first homology, but still possible to actually work with.
I agree with the other contributors here that the condition may be a little artificial from a category theory perspective, but it still may have some practical use.
EDIT:  I have since found a little on this topic under the subject of graph congruences.  A graph congruence is essentially an equivalence relation on the set of paths in a directed graph, which respects endpoints and composition of paths.  i.e. the set of equivalence classes of paths forms a category.
I'm not really familiar enough with the terms used in the papers I've found (languages, varieties etc.) to say anything about them, but see:
Arkadev Chattopadhyay and Denis Therien
Locally Commutative Categories
and its references.
