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.

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.