I haven't looked that hard, but it's kind of surprising that as far as I can see, this question hasn't come up before on meta.
MathOverflow's name reflects our proud history. Although we are hosted on the Stack Exchange network, we were founded independently and continue to be an independent entity. I'm pretty sure we pre-date the existence of Mathematics Stack Exchange, for instance. We've had this name, and the associated logo and so forth, since 2009.
None of this is precisely relevant to the valid concerns you bring up about the regular opportunities for confusion about the purpose of the site suggested by the name. But what I'm trying to express is that I, and I'm certain many others, would be extremely reluctant to change the name of the site after all these years.
Moreover, my sense is that almost any conceivable name would have similar issues. The name and setup of the site need to communicate that
We're about mathematics;
You can ask questions and post answers here;
We're not about any mathematics questions, but specifically about research-level mathematics.
I think the name and setup do a great job of communicating (1) and (2). The difficulty is communicating (3). Part of the problem is that the majority of people looking for answers to mathematical questions on the internet don't even have any idea that there is such a thing as research-level mathematics! Even if they do possess such a concept, the idea that there's a sleek, modern Q&A site devoted to something so niche and that that's what they've stumbled across it is probably quite surprising. In fact, it's an open problem, even after all these years, to adequately communicate (3) in one or a few sentences, and we continue to have internal debate about it -- for just one recent example, see discussion here. Given these difficulties, the challenge of communicating all of (1),(2), and (3) in a single name seems beyond the reach of current methods.
So if we were to try to deter off-topic questions by changing the name, the only avenue I could imagine at this point would be to choose some very clinical and stuffy-sounding name which would be indecipherable to most people on the internet, so that they would not suspect we're talking about mathematics here. I, and I think many others here, would be very unhappy with such an approach for many reasons. Among them would be sentimental value of the current and historical name, as well as worries about attracting users whom we do want to attract.
For these reasons, the name MathOverflow is not going anywhere. Phew!