I gave a huge data dump yesterday that pertains to picking every level of the bracket. Today, let's just focus on the national title contenders, and again, the trends discussed below only apply to the 64-team bracket era of 1985-present.
Setting the Stage
The initial pool of potential champions contains the teams of the first three lines of the bracket. Only three times has anyone seeded worse than a 3-seed won the whole thing. I was tempted to exclude 3-seeds themselves, as they've won just three tournaments, but two of those three have happened in the last ten years. They're in play.
Initial Pool: Ohio State, Kansas, Pittsburgh, Duke, North Carolina, Notre Dame, Florida, San Diego State, Syracuse, Purdue, BYU, UConn
First Cut: Mid-Majors
Only two mid-majors have won the national championship: Louisville in 1986, and UNLV in 1990. I'm not even sure UL counted as one at the time, though. I don't think the Metro Conference was considered a major one back then, but it was Denny Crum's second national title.
Long story short: mid-majors aren't good bets to win the national title despite Butler's close call last year.
Cut: San Diego State, BYU
Remaining: Ohio State, Kansas, Pittsburgh, Duke, North Carolina, Notre Dame, Florida, Syracuse, Purdue, UConn
Second Cut: Coaches with no Final Four experience
As Jon Solomon pointed out, 10 of the last 11 and 20 of the 26 total champions had coaches who had been to the Final Four prior to winning it all. It's not common for coaches to win it all the first time they break through to the national semifinals.
Cut: Pittsburgh, Notre Dame, BYU (again), Purdue
Remaining: Ohio State, Kansas, Duke, North Carolina, Florida, Syracuse, UConn
Third cut: Conference tournament results
No one has won the national title without winning a game in their conference tournament. Also, no one has won the national title after playing more than three days in a row in their conference tournament.
Cut: Pittsburgh (again), Purdue (again), UConn
Remaining: Ohio State, Kansas, Duke, North Carolina, Florida, Syracuse
Fourth cut: Non-conference losses
Only three teams have won the national title after absorbing more than two non-conference losses. If you're good enough to win it all, you generally clean up in the non-conference.
Cut: North Carolina, Florida
Remaining: Ohio State, Kansas, Duke, Syracuse
That's as far as I can take things for now. If you're looking for a reason to eliminate Syracuse, use the fact that only three 3-seeds have ever won it all (though one of them was Syracuse in '03). If you're looking for a reason to eliminate Duke, make it the paucity of repeat champions. Only 1991-92 Duke and 2006-07 Florida have repeated as champions, and the other two champs to even make it to the title game the next year ('94 Arkansas and '97 Kentucky) failed to win it.
As for Ohio State and Kanas, I can't think of a solid reason to cut them at the moment. Those two really are your most likely championship candidates.