Not sure if this is a GQ or a CS question, so mods please move if I put this in the wrong spot. Syrcacuse is the NCAA champ. Okay, so if you’re Manhattan, you say “We’re not so bad, we got beat in the first round but it was to the eventual champion.” Not so for Dayton. They lost to Tulsa, who lost to Wisconsin, who lost to Kentucky, who lost to Marquette, who lost to Kansas, who lost to Syracuse. Is there a name for the team that not only loses the first round but is at the head of a chain of losers all the way through the bracket? Just as there is only one winner, there is only one such team per year. And if we went back through the years, would there be any perennial winners of that dubious honor?
No, there is no name, other than the fact that the college who lost in the first round to the eventual winner can take some solice in it.
BTW, I work at Siena College, which lost last year in the first round to Maryland, the NCAA champ. And they’re also in the same conference as Manhattan, meaning the MAAC conference champ lost to the eventually chamption twice in a row. People will be lining up next year to get a chance to play the MAAC champ. 
Never heard it mentioned in terms of college basketball, but I have heard of the “daisy-chain national champion” jokingly refered to in college football, if the national champ finishes the year with 1 loss (and perhaps isn’t the clear-cut no complaints champ).
Theoretical example… The Champion Miami Hurricanes lose midseason to the Florida Gators, who lost to the Auburn Tigers, who lost to Troy State, hence Troy State is the true national champ!
Speaking of first round losers, I picked Syracuse to lose in the first round. Oh well.
Kinda related–the last person chosen in the NFL draft each year wins the “Mr. Irrelevant” award. I forgot who actually came up with this.
In breeding this line is called tail male or tail female according to the side of the pedigree, so in sport we would have tail winner and tail loser.