One advice I hear often is: if you're not the quickest, just don't try to race anyone else for solving a problem (there are many people you could legitimately call 'genius' working on certain problems!). Work on things you know no one else is working with -- if you fail, it's only because the results aren't very useful (but for most things it's really hard to tell when they'll be useful anyway). I call this "work orthogonalization".
And there's my favorite, and probably most often said: be guided by beauty. If you feel what you're solving is "just right", it most likely will work and will work well.
Common sense stuff, so it has exceptions of course. Engineering for instance has a whole extra layer where we have to deal with physics, resource constraints and follow more closely the needs and demands of our users.
I have heard that too. I was talking with a noted probabilist, who co-authored a well-regarded book, about some work that Michel Talagrand (a very accomplished mathematician) was starting to publish.
He said, "just have to get out of the way," and when I gave a quizzical look, continued, "That guy is a bulldozer." Basically, you might sweat for a year and end up being a special case of a more general result that Talagrand just proved.
Thurston's essay is another consideration of this question (http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...).
Sometimes I wonder, though, what's the purpose of associating oneself with advances in the world? If you knew other people could do it better, would you still want your solution to win? I guess the answer is that we don't know other solutions are definitely better, when everyone is developing theirs, and betteris a relative term.
Perhaps it's more interesting to think in terms of Alan Watts' philosophy of growing and competing. Things just happen and there is no "ceramic theory" of good ideas.