The pentagon and hexagon axioms in the definition of a symmetric monoidal category are one example that I was thinking of here; the axioms of a weak 2-category are another. I understand that it can be checked laboriously that these few coherence axioms are sufficient to show, e.g. in the first case, that all coherence conditions we want on associativity and commutativity to hold do, but this is rather tedious. Is there some other motivation for the choice of coherence axioms?



Motivation for coherence axioms
I agree with Viritrilbia: there is in general no "nice" or canonical choice of coherence axioms. By "in general" I mean for an arbitrary 2-dimensional theory, such as the theory of weak 2-categories, monoidal categories, braided monoidal categories with duals, etc. This is true for the same reason that there is in general no "nice" or canonical presentation of a given group.


Nevertheless, there are some famous patterns in the coherence axioms for certain commonly encountered theories. The associahedra, or Stasheff polytopes, give the higher associativity coherence axioms. (There is one such polytope of each dimension. The pentagon is the associahedron of dimension 2.) The hexagon in the definition of symmetric monoidal category is also part of a general pattern. 


Topologists often sweep units under the carpet. In the definition of monoidal category, the pentagon is not the only coherence axiom: there is also the triangle, which has to do with the unit coherence isomorphisms $X \otimes I \to X$. People talk much less about the triangle, even though it's a crucial part of the definition. I don't know of a general pattern that it fits into.


Incidentally, when the definition of monoidal category was first given (by Mac Lane?), there were one or two extra coherence axioms that were later shown to be redundant. This is an indication of how non-obvious the choice of axioms is.


