Choose a topic where you can do something computational.
Your options inside or outside the academy will be greater if you have computational skills. Most graduate students in math do not end up at research universities.
If you want to make progress on old problems, you'll probably need tools that the old mathematicians didn't use. Computers are great tools.
Getting people interested in your proofs is hard, but numbers and computational results are great bait for getting others to collaborate.
For instance, consider Schanuel's conjecture in the form:
If there are $n+1$ algebraic dependencies among $x_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n}$, then there is a $Q$-linear dependence among $x_1,\ldots,x_n$.
For which $n$ and which algebraic dependencies can you prove this? Can you devise an algorithm, and what numerical results does it give?