I. LOVE. WEIRD. GAMES. Let's go over some favorites:
The Katamari Damacy series: The stars have been destroyed (usually through the capriousness of the King of All Cosmos). You, as the Prince, take a sticky ball to Earth to roll up stuff to make new stars. Start off rolling up thumbtacks and discarded candies, end by rolling up THE EARTH ITSELF! Amazing soundtrack.
One Finger Death Punch: Stick figures engaging in DDR-style combat. Lots of blood!
Patapon: Control an army of eyeballs with the beat of your mighty war drum. Hunt, party, fight!
Locoroco: You are the planet, attempting to help adorable singing blobs avoid evil stringy blobs by altering your own position in space.
Odama: Voice-controlled strategy RPG/pinball crossover set in feudal Japan. Interesting, but too frustrating.
Oudendan/Elite Beat Agents: Solve people's problems with the power of DANCE! (Techincally a Japanese form of cheerleading, but it doesn't translate well.)
I'm sure I'll think of more later.