[Verginius Rufus]

Hillary can't pick Bill for two reasons--no one ineligible to be President can be elected VP, and they are both legal residents of New York, so the NY electors would not be able to vote for both of them in the electoral college.

The only way Bill could become President again would be if he was Speaker of the House and both the Presidency and Vice Presidency became vacant at the same time. I believe the House of Representatives is permitted to select as Speaker someone who isn't a member of the body, but I don't that has ever actually happened.

[Revenge of Sith]

If the speaker of the House (or the president pro tempore of the Senate or a Cabinet member) is ineligible to be president, then the line of succession would skip to the person below him. For example, Henry Kissinger could not have succeeded to the presidency when he was secretary of state.