Definately the island, and whichever one was as large as I could afford with half a billion.
I'd adopt the US Constitution, without income tax amendment. Except rephrased in modern terms, and EXPLICITLY STATED that no Bill of Rights Amendment may ever be repealed. And change the second to "The right of the people to own guns will not be violated, infringed, regulated, controlled, taxed, or in any other way limited." Oh, and get rid of the commerce clause since there won't be any states in the first place. In fact, get rid of congress altogether and use direct democracy. Nothing like a government where the people themselves, not a seperate representative body, have equal say in government as a president and court system.
Business would build the needed infrastructure (I'd convince them it's an investment) then the people would pay boatloads to get to the island.
But what would I name it? That's the tricky part.