Wow... hmm... not easy.
2004, no 9/11, no Iraq war. Howard Dean's campaign is a non-starter. John Edwards is more openly exposed as a philanderer and a hypocrite for the couple of dozen people to whom it wasn't immediately obvious.
This leaves John Kerry, running his campaign mostly as OTL, only with less of an emphasis on foreign policy - his plan is to win the Presidency by "not being George W. Bush"
Dick Gephardt also has more of a following now, but with the reasonable popularity of Bush is trying to position himself as a moderate. It is proving to be a hard sell though, as the economy is doing reasonably well and independents are leaning toward reelecting Bush. He succeeds in taking a fair share of conservative Democrats away from Kerry.
Kucinich meanwhile is the only openly proud liberal in the group. Many other candidates, including Al Gore were driven away due to perceived invincibility of Bush in a good economy without any controversial foreign policy decisions ITTL.
Frustrated members of the hard left flock to Kucinich though. He is the hero of Moveon.org and DailyKos. These people still believe that Bush stole the 2000 election and are excited about having their chance to be rid of him. With no Howard Dean, John Edwards, Al Gore, and Kerry and Gephardt looking like they sold out, Dennis Kucinich is their man. Republicans and moderates find this odd, but the ideologue will win the nomination this year.
Kucinich uses his midwestern roots to edge out Gephardt in Iowa. Kerry wins New Hampshire, but not strongly - Kucinich places a close 2nd. Gephardt takes South Carolina. Kucinich wins narrowly in Michigan and Wisconsin, big in Washington, Hawaii, and D.C. The other early states are split between Gephardt and Kerry as they each take a share of the other's voters. With momentum showing Kucinich can realistically take this, liberals get excited giving him big Super Tuesday victories in CA, CT, MD, MA, NY, RI, and of course Ohio.
Okay, so that gets him the nomination. Just gotta figure out what ASB scandal gets Bush so that this doesn't end up being a 1984 style drubbing.