Easy.

Usually you have a time period to claim the winnings, on the order of months.

Find a famous person from your state, ideally from the same region. Doesn't have to be a movie star. Bonus points for a gender neutral name, but not required. Try to pick someone with an ordinary name (Arnold Schwarzenegger won't work). Examples:

If you're from Florida, especially Central Florida, choose Casey Anthony.

If you're from Oklahoma, Brad Pitt is a good pick.

If you're in LA, you can pick almost any movie star since they all live there.

So when you win, you file a name change as far as fucking possible from the place where you win (i.e. another state), using the name you selected. Then you wait as long as possible to collect. Then you collect, under that name.

The reasoning for this is that Googling "Brad Pitt Oklahoma" and "Casey Anthony Florida" is going to fuck up the overwhelming majority of would-be stalkers and kidnappers. By choosing someone with an ordinary name, you increase the chance that the change will be approved. Gender neutral helps obscure your gender which is another layer of anonymity.