The other answers don't seem to have said much about why the axiom of choice is widely regarded as plausible. Let me try to address that question.

First let's dispose of some non-reasons. In response to your questions, I don't know of anyone who thinks that the Banach–Tarski paradox is a reason to believe in the axiom of choice. I also don't know of anyone who argues, "It is a priori plausible that there exist non-measurable sets; so the fact that the axiom of choice yields the attractive conclusion that there are non-measurable sets is a point in favor of believing the axiom of choice." Instead, those who are comfortable with the existence of non-measurable sets typically start by accepting the axiom of choice, and then they accept non-measurable sets as "part of the territory" that comes with the axiom of choice.

Those who think that Banach–Tarski casts doubt on the axiom of choice typically have a philosophical predisposition that *math is supposed to model the physical world closely*. So for example, $\mathbb R^3$ is not just a random mathematical structure that we study purely for its own sake; it is supposed to be a decent model of physical space (or at least, open subsets of $\mathbb R^3$ are supposed to model localized regions of physical space). Banach–Tarski, when given a direct physical interpretation in this way, yields something that we "know" makes no physical sense, and so if we think that math is supposed to yield physical truth in this way, then Banach–Tarski is going to lead us to reject something in the math. Whether that "something" we reject is the axiom of choice is a separate question, and Andrej Bauer's excellent answer shows that there are other options, but the point I want to highlight is that we're going to be led down this path in the first place only if we have certain presuppositions about how math and physics are supposed to relate.

There are others who don't view set theory in this way. According to them, set theory is supposed to be about *abstract collections of things*, and the way to arrive at axioms is by abstractly thinking about what properties they should have, not by comparing them with the physical world. The axiom of choice can be thought of as saying that if you have a bunch of nonempty collections of things, then there is another collection of things that contains one element from each of your original nonempty collections. Stated this way, the principle sounds intuitively plausible, and I would argue that this intuitive plausibility is, at least implicitly, the main argument in the minds of most people who accept the axiom of choice. If this is the way you think, then non-measurable sets and Banach–Tarski are not going to dissuade you from accepting the axiom of choice. Those phenomena will just lead you to say that we can't arrive at physical predictions from mathematics in such a naive manner; instead, to do physics, we have to formulate *physical theories*. Math can of course help a lot with the construction of physical theories, but it's not as simple as just saying that the mathematical theory of $\mathbb R^3$ *is* our theory of physical space.

These two options aren't the only options. The work of Solovay shows that you can, to a large extent, have your cake and eat it too, by working in a set-theoretic universe where all subsets of $\mathbb R^n$ are Lebesgue-measurable and a weakened, but still quite strong, version of the axiom of choice known as "dependent choice" is available. Why Solovay's model hasn't become more popular is not completely clear, but perhaps part of the reason is that it feels like a "compromise position," and the people in the two different camps above have not seen any need to migrate to that kind of compromise.

mathematicalobjects, not physical ones. That those mathematical objects happen to be bounded sets in $\mathbb{R}^3$ and that we can also physically conceive of some bounded sets in $\mathbb{R}^3$, does not mean that these particular sets have any relevant physical interpretation; and that lack of interpretation is (again, to me) completely independent of AC. $\endgroup$ – Steven Stadnicki Jan 20 '17 at 5:26