Thanks to another HN comment, I recently purchased the book
Where Good Ideas Come From: The Natural History of Innovation in order to answer just such a question. I'll leave that recommendation there, but I will admit that I haven't yet had a chance to crack the book--been too busy on my December project. I figure that's a pretty good excuse though, so I'm letting myself off the hook for awhile.
The most common and useful answer I've ever gotten for this type of a question is to simply find the smallest possible problem that you could solve, and solve it. I personally find it easiest to solve a problem from which I am suffering. This has a couple nice advantages, primarily that I can usually tell if I solved the problem sufficiently or not, and also if it flops, at least I've saved myself some trouble.
I launched http://sleepyti.me on November 8th because I personally always wake up tired. I didn't really expect much out of it, but I'm averaging about 10k visitors per day and between $15-$20 in Adsense revenue per day. Maybe not "successful startup" money, but for a free webapp that only took me a few days to write up (and only costs $20/month to host on Linode, along with all my other sites/shells), I consider it a pretty big success.
I don't think that there's a secret that can foster brilliant, successful ideas, otherwise we'd all use it and be billionaires. For me anyway, it's just the tried and true combination of persistence and solving problems.
Good luck!