Heh...
I don't know where to start. Every so often I hear this from friends down there asking about the "Move to Alaska Incentive."
So... ok. There is a Permanent Fund Dividend that's paid out to Alaskan residents on a yearly basis. This is not something you get paid just for moving here, you have to have been an Alaskan resident for an entire calendar year (that means if you move here in the summer you won't be eligible for another year, because that year won't count). And you have to be planning to stay.
It doesn't pay "thousands" of dollars... last year was a $1654 dividend. And I guess the money doesn't hurt, but don't think you're going to be getting rich off it either... I like to save my check for a snowy ("rainy") day or to cover cold weather gear, and don't think you'll get away with a windbreaker or two up here, hah! Generally you need to apply for your check for the previous year around January, and you can expect your dividend to be cut for you in October.
So it's not like you can just move up here and get money. It'll be awhile before you see anything. But it can help out a lot with the expenses if you were already thinking of moving up here anyway.
The money also isn't from the feds, it's handled by the state. You still have to pay taxes and everything else.
There's a great page run by the state gov that's located here:
http://www.pfd.state.ak.us/Snowed In (Again)