Inspired by the "What if the 1860 US presidential election was thrown to Congress?" thread.
Scenario: Lincoln loses California, Illinois, Indiana and the rest of New Jersey's electoral votes to Douglas, as well as Oregon to Breckinridge. No one now gets an electoral-vote majority, so the election is thrown to Congress. Not only that, but Douglas has now overtaken Bell to reach third place in the electoral count, meaning that he is now eligible to be elected by the House but Bell is not.
The result? Joseph Lane is easily elected Vice-President by the Democratic-controlled Senate. Meanwhile, in the House, the non-Republican ex-Whig delegations (Bell's support base) force a deadlock: naturally they won't vote for Lincoln, but being closely associated with the Unionist Party they see that they will control the balance of power in the House in the ensuing term and they'd rather do it with as weak and illegitimate-seeming a Democratic president as possible -- thus they refuse to vote for Breckinridge either, to prevent the House from electing anyone. Eventually the clock runs out and, on March 4th 1861, Vice-President-elect Joseph Lane is sworn in as the 16th President of the United States.