One indication is the qualifying exams of my students:
I don't have the exam for Megumi Harada PhD 2003.
At one point, you could get a good idea of the sort of things you would learn from me from this.
Core competencies: compact Lie groups and their actions on symplectic and algebraic varieties. Toric varieties. Side interests: Gr\"obner bases, SAGBI bases, combinatorics of Schubert polynomials. If you're intrigued by these things then I have problems for you.
False competencies: combinatorics in general, algebraic geometry in general. I use these things daily but I do not want to guide a thesis on them. Given the strength of the Berkeley department in algebraic geometry, I don't have much fear that someone would carelessly fall into an algebraic geometry thesis under me.
Combinatorics is a different story. My great worry about combinatorics is that it's too easy to end up working on a (perhaps very difficult) problem for a few years, emerge at the end with an answer (if one's lucky!), but then not be able to interest any other mathematicians in it, nor have the tools to work on any other problem. So be warned: if you come to me wanting to work on the combinatorics of e.g. honeycombs, I will force you to learn all the representation theory/symplectic/algebraic geometry that they involve.
1. The advisor should know where the easy, but important, mathematical
problems are. It may be hard to believe but at any time
2. The advisor should know the connections of the student's problem to other areas of mathematics. And now we get back to me: for lack of a better definition, I tend to define the importance of a problem in terms of the level of connection with other mathematics -- particularly other important mathematics!
While working on such problems is a great bother, in that it requires one to be abreast of many fields, the payoff is incredible: one can actually talk to people. If you can't explain what you're working on to your grandmother, it's regrettable. If you can't explain what you're working on to a math professor, it's very sad indeed.
3. The advisor should know what other people are working on. Few experiences are more frustrating than working for years on a problem, to have someone else solve it; one of them is to find out that it was already solved long ago!
4. The advisor should let other people know what his students are working on. Thankfully mathematicians are loth to work on a problem if they know some grad student has it as a thesis problem (remember: there's so much other easy mathematics out there), but it's the advisor's job to let the world know what their students are up to.
What, then, is not the advisor's job?
1. The advisor doesn't have to be correct about mathematics. It'd be nice to be able to trust someone about what's true, but this is mathematics and the only thing that counts is proof. If you don't believe my proof, that's probably because it's wrong, and you'd better not count on it. (Conversely, if the final thesis contains errors, it's the fault of both student and advisor.)
1. It should be flexible enough that partial answers are still interesting, and it's never really done - there are always avenues for further work. Problems of the form "Is the following conjecture true, yes or no?" are really dangerous, and I don't want to subject a student to such dangers unnecessarily.
2. To work on it, the student should be forced to learn a big machine, that will serve them well for other problems. Problems that one can get started on one's first day of grad school are unlikely to be important.
3. It should be possible to give a math colloquium talk, not on the problem itself, but just on the background necessary and the statement of the problem, such that the audience is then eager to hear about the solution. This may seem counter to #2, but it's not: #2 says that the problem should be related to deep and interesting mathematics, which means that in #3 you can get other mathematicians interested.
4. It should admit experiments. This usually means that one wants to prove something for all n, and can actually work out the n=2 and n=3 cases by hand to start getting ideas. Or for all symplectic manifolds, so one starts with toric varieties. Or for all groups, so one starts with GL(n). At some point the student should be forced to learn how to compute things that noone's ever computed before.