One of my players was good at coming up with odd character concepts. Like Frat Man (and his faithful sidekick, Keg-Boy, aka the Boy Kegger), who gets stronger the more he drinks. He also wanted to create a Fantasy Hero wizard with a spell: Exploding Porcupine. (Spell component: one live porcupine) And I swear that he came up with The Waffler before the Mystery Men movie came out.
Another player suggested each player create a character with points in a car (specifically the Yugo). Then, when needed, we could combine to form Yugotron, a giant car-robot.
As GM, I can't think of any character concept too goofy to at least get mention in the news during my Champions campaign, if not actually put in an in-game appearance. My players had to fight the Recycler, who built a giant robot from trashed robots and other junk left over from various super-fights. Though not my own concept, my players fought beside the man-chicken hero Rhode Island Red twice (and still think he's awesome).
I got a lot of use out of ROVer, a man with a variety of remotely operated vehicles (ROVs), most of which were designed to look like dogs.
Not a character concept, but a hero team name that might have seemed unwise: the Virginia Alliance for Global Interdiction and National Assistance. They came about because my players were dragging their feet at picking a superhero team name. So one day the news mentioned VAGINA, who were named by the editor of the Heronet Herald after a group of Virginia-based heroes refused to name themselves. (It worked wonders -- that night, the players picked a team name.)