In ZFC, there are three particular axioms that are less obvious than the others: regularity, replacement, and choice. (Replacement is an axiom scheme, but we can ignore that difference for this purpose).
Of these, regularity (well foundedness of $\in$) is the easiest to deal with. Although there is no reason to think that our naive conception of sets eliminates the possibility that there is a set which is a member of itself, it also turns out that we essentially never construct such sets in the course of ordinary mathematics. Thus the axiom of regularity does little harm (in removing things that we care about). It does do some good, as well-founded models of set theory are much more convenient to study. Most mathematicians never think about it.
The axiom of replacement is odd because it is hard to motivate directly from the notion of the cumulative hierarchy; replacement is essentially about the length of the ordinals rather than about which sets exist at each level of the cumulative hierarchy. There are very few mathematical arguments outside of set theory that actually use this axiom, though. The main examples are the Borel determinacy theorem and some theorems from category theory. Thus most mathematicians rarely notice it, it is not mentioned in many undergraduate books outside set theory, and except for set theorists I expect few would be able to state it without thought.
The axiom of choice is odd because it is a set-existence principle, but as t.b. says in a comment it is not implied by the other set-existence scheme in ZFC, the separation scheme. Unlike replacement, though, the axiom of choice can be motivated from the naive construction of the cumulative hierarchy. Historically, the axiom of choice was a flashpoint for certain discussions about constructiveness in mathematics, and for this reason, many authors in the past marked results that used the axiom of choice so that it was clear when it was used. This habit has decreased over time as the arguments from the early 20th century have faded somewhat into history; a side-effect of the habit is that it reinforced the lingering idea that there was something unique about the axiom of choice compared to the rest of ZFC.
All three of these axioms (regularity, replacement, choice) are, at various times, separated off from the rest of ZFC, leaving behind weaker set theories. The main reason that people think of the axiom of choice as special, rather than regularity or replacement, is that the axiom of choice has been the one that is most discussed in popularizations and undergraduate textbooks. But from the point of view of ZFC it is not at all the only axiom that requires effort to motivate.