I’m simply astonished that anybody would turn down a million bucks.
There are some things in this world that are worth more than money, and integrity is one of them, at least to some people.
Kind of an old story now, but critics are now saying that Perelman should have taken the money and given it to charities.
No pleasing some people.
I’m of the opinion that the guy has a form of OCD or something somewhere on the spectrum of what is autism. That would both account for his ability to solve the problem and his reaction to the prize money.
I have OCD, and, despite hating the extremely high value everyone gives money, I still would not turn down that much money, even if I didn’t think I deserved it. I would feel like it was being wasted otherwise. And, like most OCDers, I hate waste.
OK. My default was to a form of autism. Let’s go with that.
Who knows, maybe a voice in his head told him the solution, and he thinks the voice deserves the prize. Except he doesn’t know who the voice is or where to find it, because it’s him, but he doesn’t recognize that.
The story I heard from a mathematician in St. Petersburg (where Perelman lives and works) is that one million dollars is enough to make life dangerous (you or a member of your family could get kidnapped for a ransom) and not nearly enough to hire full-time protection. One million dollars is the opposite of the sweet spot. Of course, he could readily get a job in the US, but he isn’t interested.
So, if he really is smart, he’s publicly turned down the prize but actually secretly met a Clay Institute flunky with a large briefcase on the Trinity Bridge on a moonless night? 
Well, he more or less did this, since the institution which awards the prize is a charitable organization, and will presumably turn the money over into one of their other projects.
He claims turning down the current prize was because he felt Hamilton did most of the work, though one wonders why he couldn’t just take the money and give half of it to Hamilton if that were the case.
He’s turned down several other prestigious mathematical awards (including the Fields medal) and their associated case prizes, often for wildly varying reasons. I’m not sure he’s really sticking up for some “principal”, or if he is, he’s not articulating what that principal is very well. I think he just likes being known as eccentric, and doesn’t really care much about the money.
There is a book called Perfect Rigor by Masha Gessen that was published recently which I read a few months ago. It’s a biography of Grigori Perelman. There are a number of reasons that explain why he’s turning down prizes. It has to do both with his mental state and with the things that have happened to him. I’d rather not summarize the book, since I don’t know how to do that without oversimplifying it. Read it if you want to understand what is going on with Perelman.
I shall at least link to the Amazon page, which includes a fairly in-depth interview with the author.
As autism is something you are born with, and some of his predilection seems to be from past events, I’m not sure it quite fits.
Here he gives a somewhat different take on the reason for turning it down:
So it seems to be a Bobby Fischer-style spat with Organised Mathematics™ as much as anything.
And like Grigori Perelman, Bobby Fischer both had mental problems and was not treated well at times by the community he tried to succeed in.
I believe that Perelman has said that he’s saved enough money from working in the US so that he doesn’t need to work again. From newspaper articles, he seems to live frugally. Also, his rejection of the Fields Medal and the prize money has a lot to do with his dim views on the state of academic mathematics.