Can be any point and click adventure game released for any platform preferably retro. However it should have the most peculiar and odd story-line to it.
Although I haven't played it yet I heard Harvester (http://store.steampowered.com/app/287020/) is pretty strange.
I played through Harvester. Easily one of the most memorable games I've ever played. The game is just creepy yet very humorous at parts. You'll hit a certain spot in the game and things just go completely haywire.
I'll throw Normality (shop around for the different voiceover versions!) and Dark Seed (did not actually play this one, but it got stuck in my mind as "weird" nevertheless) into the ring. And Maniac Mansion, if that isn't too mainstream?
Give "The Lost Crown" a try. There's a newer version available on Steam for 10 Euro. While it may not be as odd as Harvester, it certainly is weird and creepy in its own way. The people you speak to give a very Silent Hill sort of vibe, meaning that they seem to act a little weird, they seem to know a lot more than they're letting on. Fantastic game, certainly one of my favorite horror games.
"I have no mouth but I must scream" is a really really disturbing point-and-click. "Phantasmagoria" is also a very disturbing one. Can't decide which one is the weirdest but they are both great !
One of my favourites, that I think is a little strange mainly due to how prevalent the story is with it's universe is The Feeble Files. Also, the main character is voiced by Robert Llewellyn (Kryton from Red Dwarf). Definitely check this one out. It's got great humour too.
