Basically, I tried thinking of a fun/somewhat plausible Wellstone lives scenario, and it spiraled out from there. Maybe I'll revisit this in a TLIAW or something sometime.
Presidents of the United States:
2001-2005: George W. Bush / Dick Cheney (Republican)
def. 2000: Al Gore/Joe Lieberman (Democratic)
def. 2004: Wesley Clark/Tom Vilsack (Democratic)
2005-2005: Dick Cheney / vacant (Republican)
2005-2009: Dick Cheney / John Danforth (Republican)
2009-2013: Paul Wellstone / Brian Schweitzer (Democratic)
def. 2008: John McCain/Joe Lieberman (Republican)
2013-present: Al Gore / Brian Schweitzer (Democratic)
def. 2012: Rudy Giuliani/Rand Paul (Republican)
def. 2016: Mike Huckabee/Kelly Ayotte (Republican)
Essentially, Paul Wellstone avoids the plane crash that killed him IOTL, and the minor butterflies from that results in a different Democratic primary come 2004, wherein the main candidates at the end are Howard Dean (who Wellstone supports) and Wesley Clark (who a majority of the Democratic establishment reluctantly supports, despite his unpopularity among progressives). Clark wins the nomination, but proves to be a liability on the campaign trail (particularly due to his inconsistency on Iraq), meaning Bush wins a slightly larger victory than OTL.
Bush is assassinated in 2005 by Vladimir Arutyunian, and Cheney nominates the uncontroversial John Danforth as his Vice President. Cheney quickly proves to be unpopular; sending more troops into Iraq, the conflict proves to be far bloodier than OTL. More notably is the beginning of a war between Israel and Iran in 2007 following the latter's attempt at developing nuclear weapons, quickly bringing the United States into another war in the region. As it becomes clear that the wars are being somewhat poorly run and resources are being stretched too thin, public opinion generally turns against them. Cheney's popularity, already low, further declines, leading to his announcement that, as expected, he will not contest the 2008 election.
The surge in anti-war sentiment and progressive protests against the administration convinces a reluctant Wellstone to run for President, at the urging of his progressive colleagues in the senate (Bernie Sanders and Russ Feingold among them), despite his worsening health (although at this point still okay given the circumstances) as a result of his multiple sclerosis. Establishment Democrats find their support divided between Obama and Clinton early in the primaries, allowing Wellstone to take advantage of the split vote and win key early states (Iowa and New Hampshire among them) in the primaries. With the momentum on his side and a far more passionate base of supporters, Wellstone is able to narrowly defeat Clinton to win the nomination following Obama's withdrawal, subsequently choosing Montana Governor Brian Schweitzer as his running mate.
Wellstone's nomination, and the perception that he's somewhat of a radical, allows McCain to take the lead in polling, despite the unpopularity of the Cheney administration. Hoping to take advantage of Wellstone's troubles with centrists, McCain taps Joe Lieberman as his running-mate to appeal to moderates. This does more harm than good, though, as the pick outrages conservatives, who instead plan to stay home on Election Day. The onset of the recession further harms McCain, allowing Wellstone to win a narrow victory. His government proves to be somewhat mediocre; although he is able wind down the wars in Afghanistan, Iran, and Iraq, increase the tax rates on the wealthy, and introduce universal health care, most of his proposals (like the latter two) are watered down by Congress or, in the case of such things as increased gun control, ignored entirely. More notably, however, his first term sees a relatively rapid decline in his health, though this is largely kept out of the public eye. With Congress being somewhat uncooperative and with his health significantly worse than in 2008 and the stresses of the job only exacerbating its decline, Wellstone announces in late 2011 that we will not run for a second-term (ultimately passing away in 2014).
The 2012 Democratic nomination is fairly hectic, but ultimately ends up coming down to Schweitzer and Clinton (still in the senate at this point) in a race noted for the personal animosity between the two candidates. The race ultimately produces a deadlocked convention, and with Clinton and Schweitzer both steadfastly refusing to support the other the convention turns to a compromise candidate - specifically, former Vice President Al Gore. Appealing to both camps, and agreeing to both keep Schweitzer on as Vice President and name Clinton as Secretary of State in the event of his victory, Gore is ultimately able to win the nomination before defeating Republican candidate Rudy Giuliani in November. Gore's first term is defined by a strong focus on environmental issues, though he annoys progressives by increasing the presence of U.S. troops in the Middle East and Northern Africa. While the economy is still struggling by the time of the 2016 election, he's ultimately able to win re-election, thanks in part to a flawed Republican candidate in the form of Mike Huckabee.