My cats have all have odd habits:
We keep a stash of plastic grocery store bags downstairs in the basement for litter scooping purposes. Just so that the cats don't suffocate playing with them, we tie them in knots. Well, Izzy is obsessed with them. She drags them up from the basement and leaves a trail of them upstairs. She's been doing this since was was still pretty small, i.e., when the bags where as big as her. The cats love to play in boxes, we we often have one or more sitting around the house. I looked in one of them the other day and there were plastic bags inside.
Izzy also likes being carried around in stuff. When we cart the laundry basket up and down the stairs, she likes to get in it and go for a ride. She also likes to be carried around in our shopping bags. She's not a mellow cat otherwise, so it cracks us up that she not only tolerates it, but seems to enjoy it.
Sophie had a two mouse toys (the one that are about three inches long with a bell at one end). She got them when she was a kitten. Nearly every night around 8 or 9 p.m., she's bring me one of the mice. As she carried it to me--often from another room--she'd make this funny cry the entire way. It was a very mouse-specific cry; she otherwise never made that sound. When we had her cremated, we included one of the mice. The other one sits next to her ashes.
Henry doesn't bring me anything on a regular basis, but he is completely and utterly obsessed with newspaper. He licks it, he lays in it, he bats it out of your hand as you try to read it, he licks it some more. He waits at the front door while you retrieve it in the morning and then follows you around until he can get to it. It's apparently all that he thinks about. He also does something we call Freakout Cat. Henry is a pretty lazy animal, even as cats go. But about once or twice a day, he goes absolutely crazy. It often starts with him running toward a door jam and jumping up about five feet. He plays with all his toys in a flurry. He makes a bunch of wacky noises and cries. He hides under a blanket and then claws at it like he's trapped. He'll do some combination of this for about five minutes. Then he stares at you and blinks his eyes for a few seconds before promptly falling asleep. It's the funniest thing.
I love cats, but they are so strange (and infinitely entertaining).