Having read Feynman's books and descriptions of him in other places it seems to me he was supremely gifted but also willing to indulge his curiosity about seeming trivial things that is uncommon in professional scientists - the whole thing with spinning plates being a good example.
There are so many supremely gifted people out there, but almost none of them reach the Feynman-ian stage of problem solving. I just remembered an extremely interesting thing I can't remember one example of his "gifts" in childhood. He messed with radios and had a personal lab, but even I did stuff analogous to that and I am certainly not "supremely gifted" or gifted at all. So did you and almost every HN reader.
His "gifts" become apparent in mid adolescence, and the interesting thing is that these are more a combination of hard work and his environment than gifts. Let's stop talking about Richard Feynman for a moment, did you know he had a sister Joan Feynman who became an extremely senior scientist at NASA? {see: http://www.popsci.com/scitech/article/2002-04/my-mother-scie...I'll put it up separately too it is worth it. [edit: It's up over here http://news.ycombinator.com/item?id=1520685]}
She was brilliant too and had the same characteristics as her brother. Now one would assume that his gifts would be unique, a genetic lottery. It is quite often that siblings of famous scientists achieve next to nothing in life compared to their brothers/sisters. However, here we have someone who fought her way up life pretty much the same way Feynman did.
Perhaps it wasn't a genetic lottery, but an environment constructed by an amazing father who taught his kids that in order to do anything in life you need to work hard at it until you fail, and then you try again.
You know it is so difficult to accept this, but Feynman the phenomena was nothing more than dollops of hard work and persistence with an open mind. No wonder he used to scoff at IQ tests. I have a higher IQ than Feynman's reported IQ and this being HN it is highly probable that the person reading this does so too, but am I "smarter" than him? No way.
However, there is one thing he had in him that most people don't have; creativity, but this too can be cultivated slowly over time...
