I did one item already on Hubert Humphrey, but I hope you won't mind a second one. Nelson W. Polsby did an edited volume of counterfactuals once, and his contribution was "What If Robert Kennedy Had Not Been Assassinated?" It's been a while since I read it, and it doesn't seem to be readily available through a bit of googling (or, alas, on my bookshelf), but the basic story Nelson told was that Humphrey would have won the nomination; would have offered the VP slot to Kennedy; and Kennedy, being among other things thoroughly a politician, would have accepted. The Democratic Party thus unified, Humphrey/Kennedy go on to defeat Nixon, and all sorts of good things (including, perhaps most importantly, a relatively quick exit from Vietnam) ensue.
That's what I remember from the essay, and I think it's all quite likely. One of Humphrey's problems in 1968 was that he was trapped by the prospect of LBJ withdrawing support; with Kennedy on his side, not only would many antiwar Democrats have been a lot more favorably disposed towards him even without any other changes, but at least in my opinion he would have had a lot more leverage to ignore Johnson. As it was, Humphrey wound up very close in the popular vote but down 301-191 in the electoral college (with Wallace taking 46 electoral votes). A three-point swing in each state would have brought Missouri, New Jersey, Ohio, Alaska, and (just barely) Illinois to Humphrey for 74 additional EV and a 265-227 lead. California, which Nixon held by 3.08%, would have put Humphrey over the top without going to the House. There's no way to prove anything one way or another, of course, and usually VPs don't make any difference at all -- but in this particular counterfactual, I think it's at least plausible that Humphrey/Bobby does three-plus points better than Humphrey/Muskie.
Now, I don't remember what Nelson speculated down the road, if anything, other than as a long-time Humphrey supporter he certainly thought HHH would be a good president. Just for fun, though...it's easy to imagine the oil shock followed by the deep 1974-1975 recession giving a somewhat younger Ronald Reagan the presidency in 1976 over Vice President Bobby Kennedy, perhaps with a GOP Congress (no Watergate landslide in 1974, for one thing). After that...well, I guess it depends on what you really think of Reagan. It's easy to argue that a pre-Laffer, younger, Reagan serves a couple of successful terms and actually establishes the GOP as a majority party for a while. On the other hand, it's also possible to imagine Reagan having a miserable time with the same things that Jimmy Carter couldn't handle, and Democrats reviving in 1978 and 1980.
But wait! I cheated a bit there; I ignored the 1972 election. I'm stipulating that Humphrey is pretty good at the presidency, and if so (and given what else we know) it's likely that he wins a major landslide in 1972. Over who? Most likely, Ronald Reagan, then halfway through his second term in California, and the clear conservative champion. It's true that in this alternative history the nomination process hadn't been reformed in the same way, but it's very hard for me to see Rockefeller or someone from the liberal wing winning, and I'm not sure who was available to succeed Nixon as the person who was tolerable to both sides. Maybe Reagan sees the writing on the wall and waits until 1976, but I doubt it. I say he runs, gets the nomination, and is totally clobbered.
And you know what? A history in which Goldwater and Reagan get clobbered (plus Nixon, originally from the conservative wing, losing two close elections) might just be enough to shift things around considerably. How? I have no idea. Perhaps it's Howard Baker or someone like that who becomes president in 1976, and Republicans evolve into a moderate conservative party with Reaganites marginalized, and wind up with a long-term majority instead of the deadlock that actually happened after 1980. Perhaps a true three party system emerges, with a southern-based, Wallacite Conservative Party replacing the Southern Democrats. Maybe then Republicans wind up relatively isolationist and libertarian, while Democrats keep serious Cold Warriors in the fold. Who knows?
I'm enjoying the comments over on the other Humphrey post, and I can't wait to see what y'all say on this one.