Obama could pick Jimmy Carter, but not Bill Clinton.
He can't pick Jennifer Granholm, but he could pick Luis Gutierrez.
And Dick Cheney didn't really have to change his registration to Wyoming in 2000 to be on the ticket, but only because Al Gore probably wouldn't have stopped him from being inaugurated.
While the speculation rages on unabated in these final days of uncertainty, let's take a moment below the fold to look at the requirements for a Vice-President under the Constitution. (And a bonus: how Bob Barr or Nancy Pelosi could become President if there's a 269-269 tie)
The most pertinent Amendment covering the election of the Vice-President is the Twelfth, which states:
The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and all persons voted for as Vice-President and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate.
The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted.
The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.[ALTERED BY 20TH AMENDMENT]
The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
Let's start at the top: "one of whom, at least, shall not be an inhabitant of the same state with themselves."
This clause, commonly called the "Habitation Clause," is not binding on the parties or the candidates, but on the electors. The basic rule is that the electors of each state have to make sure that, when they cast their ballots for President and Vice-President, at least one of the names they select is from outside the state.
In practical terms, that means that the Vice-President and the President can be from the same state, but that state--and only that state--would have to vote for someone else to be Vice-President.
So let's say Obama decided, for example, to select Congressman Luis Gutierrez of Chicago for his running mate. Every state's electors except Illinois's could cast their ballots for Obama/Gutierrez, and Illinois would have to cast theirs for Obama/Someone Else. As long as Obama/Gutierrez win enough electoral votes for Gutierrez to pass 270 without Illinois's delegation, he's in.
Now, suppose that Obama wins more than 270 electoral votes, but Gutierrez wins less than that. Since Obama won the election, it's almost impossible to envision McCain's running mate winning 270 votes himself, which would mean there is no majority choice for Vice-President. Under the 12th Amendment, the choice would then fall to the US Senate. And the Senate, being majority Democratic, could then select Gutierrez.
So technically speaking, back in 2000, Dick Cheney could have remained a citizen of Texas and still been Bush's running mate. He wouldn't have had the requisite number of electors to be elected Vice-President on Election Day, so his name and Joe Lieberman's (as the top two finishers for VP) would have been given to the Senate, which was, from January 3rd to January 20th, controlled by the Democrats with Vice-President Gore casting the tie-breaking vote. If he forced the issue, he could have made Lieberman the Vice-President, but I think it's highly unlikely that he would have done so (given that he instructed Democratic Senators not to sign on to the protests of House members over the election). So Cheney likely would have become the Vice-President anyway.
As to why Jimmy Carter but not Bill Clinton could be Vice-President, take a look at the last sentence of the 12th Amendment: "But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."
There's actually a very interesting Constitutional argument one could make here. According to the 22nd Amendment, which set term limits on the Presidency:
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.
The ambiguity in that passage is critical: since the Vice-President would only assume the office of President and not be elected to it, could someone who has served two terms as President be eligible to serve as Vice-President, knowing that they might have to re-take the top spot? I think the practical considerations of the question mean that no, a two-term President could not serve as Vice-President, because there is no mechanism--other than impeachment and removal--that could require someone to surrender the office of President mid-term. So suppose (God forbid) Obama was struck with a terminal illness and died a year into his term with Bill Clinton as his Vice-President: Clinton would then become President, but he would become ineligible to serve almost a year before the next election. However, there's no way to compel Congress to remove him if he doesn't resign himself. That situation would allow someone to exploit a loophole in oversight to subvert the Constitution, which I don't think is an acceptable risk.
But on the other hand, Carter was only elected once, and could serve a term and a half without violating the 22nd Amendment. Just like Reagan considered Gerald Ford for his Vice-President, so too could Obama consider Jimmy Carter for his.
Finally, the bonus: in the event that no Presidential candidate receives a majority of the Electoral College, the election is thrown to the House to decide, with the Members voting by State. But the names they may consider are not the top TWO finishers, but the top THREE. In this election, that will most likely be Obama/McCain/Barr (though it could be Obama/McCain/Nader, or, heaven help us, Obama/McCain/McKinney). The idea behind sending three names instead of two (as is done with the Vice-Presidential contest) seems to be to provide Congress with a compromise option in the event of a two-faction deadlock. Technically speaking, the 20th Amendment provides that, in the event of a deadlock lasting past Inauguration Day, Congress may "by law" name someone to act as President until the dispute is resolved, but that vote might be subject to filibuster (I don't know if the Senate rules would prohibit such a move, and it's quite possible that they never considered the matter and never made any provisions to that effect). More likely is that they'd allow the Presidential Succession Act of 1947 take effect, which passes the baton to the Speaker of the House, whom I presume would still be Nancy Pelosi. But on the off-chance that enough people refuse to support Pelosi, and the Senate Republicans (assuming there are still enough of them to pull off a filibuster) decide to block a concurrent resolution naming an acting President, then the job may just fall to the only other person being considered by the House: Bob Barr. A nightmarish, but still Constitutionally possible, scenario.
So as we build our speculative frenzy to a feverish climax, let's keep in mind that the list is larger of potential Veeps on both sides is larger than we might have thought, and the ways in which this could go to Hell in a handbasket are myriad.
Sweet dreams.