Anthony Grist's answer completely answers this, I recommend you upvote his instead. I'd forgotten the exact way Sirius gives his permission (dummy David, how the hell would Sirius have gotten one of the forms anyway!?).
Good question! Since there isn't any canon explanation, all we can really do is speculate on the possible answers to this problem.
1) Oversight by JK.
Unlikely, considering both her track record and the fact that this isn't a minor detail that slipped by, it's an actual plot point.
2) McGonagall knew about Sirius.
I find this rather unlikely. Considering her confusion at the order to go fetch the large black dog at the end of the next book, she seems unaware of Sirius Animagus state. Without knowing this bit of info it's hard to imagine she figured anything out on her own at the end of POA. And it seems very unlikely Dumbledore could explain to her the situation in any satisfactory way without mentioning this fact, since it's crucial to the narrative of his innocence.
3) McGonagall doesn't check WHO signed, just that it's signed.
For any normal student I'd consider this the most likely option, since these forms are really just a liability thing. But in this specific case, it's not only Harry Potter, but the same Harry Potter who complained last year about not being able to get a signature for Hogsmeade during all that commotion with his murderous godfather. Seems quite likely she'd be curious who signed it this year. Maybe you could get round this by having the forms magically sealed so that anyone who signs MUST be a parent or guardian, so the teachers really wouldn't need to check?
4) Dumbledore intervened somehow
By far the most likely answer. Remember that Dumbledore is in contact with Sirius all summer, so he likely knows about the Hogsmeade permission. Perhaps he directly intercepted the permission form, or perhaps he told McGonagall that he himself got Sirius to sign it while he was talking with Sirius at the end of POA (as a kind of penitence thing or something). The most important thing here is that McGonagall trusts Dumbledore above all others, including the Ministry of Magic. If he acts shifty and tells her to overlook something, it's most likely she will.
These are the options I can see, maybe there are others. Of course, if anyone has any quotes from JK or anything that would certainly solve the problem!