The Alaska State Lottery:
To qualify you must be a resident of the State for one year covering October to October ( If you arived in November 08 you will have to wait till October of 2010 to qualify). You also need legal residency in the US (Green Card).
The last year I was up there(2004) the payment was $1100 per individual and Synthia was right, the amount is reducing each year.

Homesteading and mining claims:
Still valid, but you have to work the claim to maintain it or occupie the land to keep it. Even in a State as big as Alaska you'll find it near impossible to get land that way without owning a plane to get you there. All the good bits were grabbed a long time ago.