I'm not sure this is appropriate, as I am remembering the answer from an undergrad class about 10 years ago, but AIM cited the Treaty of Fort Laramie in order to seize the land. Basically, the treaty (signed in the mid 1800s) stated that any Native land acquisitions that either stopped being used by the Federal government or fell into disrepair would be re-appropriated for Native American use. The US government, for multiple reasons, decided to shut down the prison on Alcatraz at that time and didn't have a replacement use for the island. Originally the island had been Native land, and AIM seized onto that and the Treaty of Fort Laramie as a jumping-off point for their movement. It was basically a high-profile maneuver that guaranteed them a spotlight, and raised a lot of awareness across the country for Native American rights.

I can try to search for the books I had on the subject, but that was a decade ago and I don't live in the same country anymore, let alone the same state, so it might be challenging :)