I wonder whether the ability to pick out interesting problems is innate or can be taught. What do you think?
I really am not sure. A professor I used to know said that taste in problems is something that can't be taught. I am not sure I agree with that completely though. I think that in grad school it is possible to train people to ask why a question should be considered interest. In fact one might argue that asking the right questions is the core of good research.
Every field has the foundational issues that drive them. In theoretical computer science, and computational geometry, one of the key issues is often: What is the key process underlying the structure of a problem, and how can this be used to solve it ? A paper that can answer this question will more often than not be deemed interesting.
So it is important to understand what the raison d'etre of your chosen field is. This is usually never stated explicitly, but there is usually a tacit broad consensus on what it is, and this is where an advisor, and good training, can help a lot.
Beyond that though, it is probably not a good idea to worry too much about what is interesting. A vibrant research field is a mosaic that comes from individual contributions in many different directions. It would be boring if everyone did the same thing. One should be willing to take the effort to sell one's contributions though, and not stand on ceremony waiting for others to realize what pearls of wisdom you have placed before them :)