Well, having Carter die in office is too easy, so here are two alternatives I have posted about in the past:
(1) Ford wins in 1976, and things go so badly for the country in 1977-1980 that nobody (including Reagan) can save the GOP brand in 1980 against any plausible Democratic candidate (including Mondale).
(2) In OTL, Mondale runs for the Senate in 1982 (Durenberger was vulnerable and IMO Mondale would have been a stronger candidate than the young Mark Dayton, despite the latter's money), skips the 1984 and 1988 presidential races (feeling that it will be hard for a Democrat to win the presidency given peace and prosperity), gets re-elected in 1988 (a good year for Democrats in the Upper Midwest) and then runs for president in 1992, sensing that GHW Bush is vulnerable. (And even if he isn't, Mondale would still after all be able to keep his Senate seat, at least until 1994.)
Why do I think he might get the nomination in 1992? Because all the other major Democratic candidates in 1992 (Clinton, Tsongas, Brown) were trying to show they were in some way or other "different" from traditional Democrats. Mondale can be "different" simply by *not* being different--by being an old-fashioned New Deal labor-liberal. (I am of course assuming that Mario Cuomo again decides not to run.) There are enough such voters in the Democratic primaries in 1992 to give him a real chance of winning against divided opposition. Remember that a Mondale who had not run against Reagan would not have any particular reputation as a "loser"; that the Carter-Mondale ticket lost so badly in 1980 would be blamed mostly on Carter.
Could he win the general election (which we'll assume will still be a three-way race)? No doubt he would be more vulnerable ideologically than Clinton but (a) he would not be vulnerable on the "character issue", and (b) he could win even if he lost every one of the Southern states Clinton carried. In fact, let's say he not only loses Arkansas (6 electoral votes), Georgia (13), Kentucky (8), Louisiana (9), and Tennessee (11) but that by doing worse in rural and small-town areas of Ohio than Clinton did in OTL, he also loses that state (21). Also have him lose four other narrow Clinton states: Nevada (4), New Hampshire (4), Montana (3) and New Jersey (15). He would still have 276 electoral votes, six more than necessary to win.
http://en.wikipedia.org/wiki/United_States_presidential_election,_1992