I believe Arizona is the only state that has that requirement, oddly enough; NPR discussed it after McCain got the nomination (Arizona has a Democratic governor). In Alaska and Mass., a vacancy is filled by a special election, IIRC. In the 47 other states, the governor has a free hand and may even (although in practical terms is unlikely to) pick someone of a different party from the person who just left the Senate. Some governors have even appointed themselves to Senate vacancies, but most of them have lost at the next election, punished by voters who resent overweening self-promotion even by previously-popular governors.
I heard somewhere yesterday that Palin could appoint herself to Ted Stevens’ seat (if he resigns).
As of 2003, Arizona, Hawaii, and Alaska required the governor to appoint someone of the same party as the outgoing senator; Utah and Wyoming restricted the governor further, requiring the governor to pick from a list of three candidates provided by the state branch of the party in question. Alaska changed their law to require special elections in 2004, but as far as I’ve been able to tell, the laws in the other four states are still on the books.
Actually, I believe she has to resign and then her Lt.-Governor would become Governor and appoint her. That’s how it was done years ago here in Minnesota.
But it would only be for 90 days or so, because they are required to hold a special election within that time. And then the voters would likely toss her out – that is the pattern with self-appointments, like Elendil’s Heir said.
More likely, she’d appoint a caretaker Senator (Mrs. Ted Stevens?) and then run for the seat herself in the special election. Or she could appoint her husband.