Terry Tao has some helpful ideas.
I can't say I've really mastered the practice of choosing problems. I guess if there's an area of math that you're really familiar with, questions will just start coming to you, but I can't tell how much of that ability comes from some osmotic process from the examples of peers.
When I'm stuck, I try to write down where I am in a lot of detail, then I take a break. If I'm lucky, I know what sort of knowledge might be able to crack my problem, and if I'm even luckier, I know who has that knowledge.
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