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Carrying a Dozen Problems
Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: ‘How did he do it? He must be a genius!’ – Anecdote from "Ten Lessons I Wish I had Been Taught" by Gian-Carlo Rota
The hard open problems have been attacked from every angle. All the obvious attempts have been made. Some persevere on the same problem for ages (Andrew Wiles worked on the proof of Fermat's Last Theorem for six years in secret before casually proving it at a conference). For the rest of us, it might be wiser to just keep a dozen problems in our heads like Feynman, continuously learning and coming across new strategies, breakthroughs, and connections.
Another related passage from Hamming's You and Your Research (my summary),
Most great scientists know many important problems. They have something between 10 and 20 important problems for which they are looking for an attack. And when they see a new idea come up, one hears them say "Well that bears on this problem." They drop all the other things and get after it. Now I can tell you a horror story that was told to me but I can't vouch for the truth of it. I was sitting in an airport talking to a friend of mine from Los Alamos about how it was lucky that the fission experiment occurred over in Europe when it did because that got us working on the atomic bomb here in the US. He said "No; at Berkeley we had gathered a bunch of data; we didn't get around to reducing it because we were building some more equipment, but if we had reduced that data we would have found fission." They had it in their hands and they didn't pursue it. They came in second!
The great scientists, when an opportunity opens up, get after it and they pursue it. They drop all other things. They get rid of other things and they get after an idea because they had already thought the thing through. Their minds are prepared; they see the opportunity and they go after it. Now of course lots of times it doesn't work out, but you don't have to hit many of them to do some great science. It's kind of easy. One of the chief tricks is to live a long time!