This is said on the Pixar Wikia:
The literal meaning of the word "Axiom" is something that is taken for granted, reflecting on the population of the Axiom's over-indulgent lifestyle.
However, as a mathematician, I'd like to offer an alternative view of the name (which may have nothing in common with the official reasons for this naming).
Axioms are basic statements of any mathematical theory, that we consider to be true (inside that particular theory) for granted, i.e., without proving them (they cannot be proven, since there are no other statements on which to base any proof). All other statements are built upon them and they are true only as long as the axioms are true.
So, in a way, axioms are origin of every (mathematical) system/theory, and I see the above name as a subtle announcement that this ship (among hundreds of thousands of spacecrafts built to keep humanity safe and happy while Earth was being rejuvenated) will be the origin, as it actually happens to be in the end (the origin from which humans repopulate the Earth).
Edit: The captain's name is B. McCrea. Is it only by accident that "McCrea B." sounds a lot like "McCrib" or is it also a suggestion of the cartoon's finale? It might seem a bit far-fetched, but maybe it's not, considering the other names in the movie. For example, VAQ-M (a vacuuming robot), PR-T ("pretty" hair dresser), L-T (a "light" bot), BIRD-E (golfer with a golfing reference name),... The list with the name explanations is here.