Pinball Quest ranks pretty high on my list of strange games that kick ass. Pinball RPG ftw.
Perhaps it's too mainstream but any Kirby game could be considered pretty offbeat if you think about it. Although if you go there then pretty much any action platform game with a "strange" lead could qualify. And most of them are pretty strange.
Batman: Return of the Joker. I think what makes it offbeat is that they took Batman, who is famous for his gun phobia, and stuck him in a Contra clone where he blasts the shit out of everything in his way, including the Joker who is piloting a giant mechanical ship thing during the final confrontation. An extremely strange application of the Batman property that turned out to be a very good game.
M.C. Kids. Even ignoring the McDonald's theme, the game is just funky with all sorts of physics defying lunacy and minor puzzle solving not typical of platformers.
Blast Works. A great shooter for the Wii that involves turning debris from the shit you blow up into parts of your own ship.