Okay. I'm not as well-researched on '04 as I'd like to be but let's try just to get a discussion going.
-John Kerry is killed in Vietnam.
A ridiculous amount of butterflies for such an unpopular presidential candidate. He was one of the leading Democratic veterans and it's hard to tell how things change without him besides opening new doors for others. The lack of a Kerry candidacy can potentially derail Obama's presidential hopes, since he was an early ally for him. If aforementioned James Shannon becomes Senator, he could have his own presidential ambitions by now.
-John Edwards is hit by a car in 1995.
I don't imagine there being many butterflies outside the national level, but even the biggest potential butterfly is knocked out of the park by everyone else dying. Maybe his wife finds someone new though.
- Howard Dean loses the Democratic Primary to be elected to the State House in 1982 (Let's say a different candidate enters)
Don't know who else becomes Governor of Vermont but whoever they are, they have the potential to become a presidential candidate themselves, so there is that.
- Dick Gephardt is drafted in Vietnam and is killed by a stray bullet.
The House Minority Leader position opens earlier and another Dem enters it. I'm not sure who, but it could elevate someone else to a similar national profile, creating butterflies. Not to mention his absence in '88 now...
- Dennis Kucinich loses election to the House of Representatives in 1996, and loses interest in politics.
He runs anyway, because he's Dennis Kucinich, but without a Wikipedia article to explain who the hell is, so he gets no votes.
- Wesley Clark gets killed in Vietnam.
Not touching this one.
- Joe Lieberman decides to retire in 2000 for some reason or another.
Lieberman's makes a gaffe and is forced to retire from the Senate to save face as a result. He considers a run in 2004 but forgoes it due to enjoying retirement. Still offers endorsements and junk though. Becomes an independent because it's a cool label when he continues defending Republican policies and Democrats turn on him.
- Bob Graham retires six years earlier.
Half-serious 'who the hell is Bob Graham?'. I know who he is, but in the context of 2000/2004 he's incredibly inconsequential and I've only ever seen him brought up as 'X needs a running mate who can win Florida. How about Bob Graham?'. IIRC he's fairly popular in his home state and all, but this isn't going to change as much as it would if he were alive.
'
- Carol Moseley Braun loses to Alan Dixon in the 1992 Senate Democratic Primary.
The state of Illinois collectively breathes a sigh of relief.
Anyway, in this case, here's the slate of potential candidates I see, if we trap national-level butterflies in a cage:
- Fmr. Vice Pres. Al Gore (TN): Well, let's be honest here, in such a narrow field, he's got to be looking pretty appealing right now. He may not be charismatic and may be awkward, but hell, he's got experience and nearly won, right?
- Sen. Bob Kerrey (NE): As mentioned by Duke 4, Kerrey is a Vietnam veteran and had strong credentials in red states, so he's a pretty strong contrast against Bush and Iraq. He's an elder statesman, and in such a weak field, he has strong potential to stand out.
- Sen. Joe Biden (DE): He's charismatic, has an anti-war record, and can be pretty damn intelligent beneath all the grins and jokes. I'm still surprised he wasn't more speculated in 2004 - he could've been a hard hitter against Bush if he gave it a shot. (I largely attribute his failure in '08 to who he was up against.)
- Fmr. Sen. Bill Bradley (NJ): As mentioned, he might dip back in with a field like this. Who's gonna beat him this time, Al G- oh. Right.
- Sen. Hillary Clinton (NY): If the Democrats have literally nobody to offer, and if there's a 'draft Hillary' movement, she might forgo her campaign promise to serve a term in New York 'for the sake of the people and her country' or something.
- Sen. Gary Hart (CO): A surprise sleeper, but he wanted to run in '04 OTL and he's still known in some circles for his foreign policy credit. He's a good campaigner and can be strong in a weak field.
- Gov. Tom Vilsack (IO): He looks like a turtle and he's a protege of a certain Iowa senator. Give him a shot.
Honestly, I'm just imagining a lot of "My country clearly needs me" sort of attitudes springing up.