Something like this happened a few years ago in Minnesota when Sen. Paul Wellstone was killed in a plane crash just days before the election. Walter Mondale ended up taking the reins, getting an official nomination from the Democratic party, but he lost pretty badly. My guess is it would work the same way - they'd nominate a new candidate. I imagine it would probably be Hillary.
Had Wellstone lived through the election and died after winning, I'm not sure what would happen. What happened in 2002 was our governor, Jesse Ventura, appointed Dean Barkley (of the Independence party) to serve out Wellstone's term.