There is something called "Metal Wolf Chaos" - it's about a battle between the President of the US and the Vice President.... and both of them wear huge combat/mech suits, and fly into out space and have lines like "Nothing is pointless.... and the reason is because I'M THE PRESIDENT OF THE GREAT UNITED STATES OF AMERICA YEEEEEAAAAAHHHH". Predictably, it's a Japanese game and was never released here.
Personally, the strangest game I've ever played was LocoRoco.
But when you think about it, a lot of games are really strange. Take the Bomber-man series. Cute little guys run around trying to kill each other with explosives. Or what about the Super Mario series? A PLUMBER goes around trying to save a princess kidnapped by a dinosaur, killing turtles that can fly, throwing fireballs and consuming mushrooms which greatly increase his size.
It depends on your definition of weird. If you are talking about absolutely-insane-makes-no-sense-whatsoever type "Weirdness", I'd go with either Killer 7 or No More Heroes.
But then again... there is the Metal Gear Series, and we all know how strange they can get. Especially MGS3 - A Man who can control bees? Check. A crazy cosmonaut with a flamethrower? Check. Lizard Guy? Check. Ghost-guy-who can summon the dead? Check. Electric guy? Check. It was just crazy. Oh, and all the Russians spoke in perfect north-american english. I know it was because they were just too damn lazy to find someone with a Russian accent, but it was weird for me. That said, the Metal Gear Solid games are still pretty good.