A pretty big assumption to make, considering a fart in a lift in Houston in September could have probably changed that election through butterflies. Edwards was an unknown at this point, I'm pretty sure he didn't develop his true populist guise until he became a presidential candidate, but considering Gore ran as a populist during that campaign (at least in part) it's not going to hurt the ticket if he is.
A lot here would depend on how Edwards performs in the debates against Cheney. If he bombs, then Gore is going to get a lot of stick. Gore is so presidential himself in terms of FP experience I don't think it matters too much if Edwards just floats it though, and the same for the inevitable criticsms of Edwards being a new boy Senator. He's there only to do what Lieberman did IOTL, which is to provide a squeaky-clean New Democrat who distances Gore from Clinton. In Edwards' case that is bolstered by him being 'fresh'.
Assuming he does OK, would two southerners being on the ticket pick up enough votes in Florida to balance whatever Lieberman being on the picked up, and then a few hundred more? Who knows.
Assuming a loss, Edwards will probably be playing it very conservatively up to 2004 as the probable front-runner. Which means he still votes for the Iraq War ITTL, and cue opening for Howard Dean. The big question leading on from this, though, is whether Edwards gets Gore's endorsement as his former running mate. If he does, then that's a big plus for Edwards, make no bones about it. (Before any smart ass brings up his endorsement of Dean in OTL, I'd point out that Dean's candidacy was largely a bubble anyway IMO, and in so far as it wasn't it was crippled for reasons other than Al Gore; Edwards is coming from a different position ITTL as an establishment candidate, and as such Gore's status with the left of the party which he picked up after 2000 would be a big boon to his candidacy.)