Well for the slightly out of the ordinary I think my favorite would be the little white mouse I had named Pipsqueak, she was the sort people buy to feed their pet snakes but when I saw those cute little pink ears and that darling little face peeking up at me from the glass cage they were kept in when I was at Petco I just had to get her.

As for the really odd, I've only had two and my favorite of those two was Squirtle the painted turtle. Not terribly odd really, but I sort of adopted him after the kids I babysat rescued him from the middle of a busy road.
They had a cat so they let me have the turtle. ^_^
My mom didn't like him much but I thought he was adorable.