If we are going by nicknames, there's no topping the UC Santa Cruz Banana Slugs. There's a Fightin' Artichokes out there somewhere too, as well as a Boll Weevils, and a Geoducks. But Banana Slugs is on a whole nother level.
Bad pun wise, the Pace Setters are No. 1. Not a fan of Stetson Hatters, but Pace Setters is worse since a setter is a breed of dog that actually could be a mascot at a school other than Pace. Nobody would ever be the Hatters anywhere else.
For just the name of the school and not the nickname, I'm always amused by Transylvania, which is in Lexington, Kentucky. My mom once visited Lexington while I was in high school and decided she had to get me a Transylvania t-shirt - and then the checkout lady in the bookstore offered me a scholarship (even though I was 1000 miles away). The "I own a College" joke was beaten to death by Jim Valvano in the 1980s.
For league-wide mascot fun, there's the NESCAC, which includes the Williams Ephmen/Purple Cows, the Amherst Lord Jeffs, the Trinity Bantams, the Bowdoin Polar Bears, the Hamilton Continentals, and the Conn College Camels. When Williams uses Purple Cows, and faces the Delaware Blue Hens, it's one of the only all-female mascot battles in the NCAA.
For mascots, you can't top the Rhode Island School of Design and "Scrotie", who was pretty much Señor Testiculo before Señor Testiculo. But even Scrotie doesn't look more ridiculous than the Stanford tree.
For mascot rivals within conferences leading to double entendres, the USC-Oregon State game has Trojans meeting Beavers (also true in local high school battles when Weaver faces Simsbury).
For lame, Syracuse and Georgetown features a boring color for a nickname (unlike Crimson or Cardinal, which has character), against a nickname deriving from a part of a chant that translates to "What Rocks?", with an accompanying bulldog for a mascot that has nothing to do with anything.
And the nickname most spoiled by a popular movie, the Utah Utes - forever now associated with My Cousin Vinny.