Arguably the Aardvark because it is the only surviving member of the order Tubilidenta so has no close relatives or the only other order with only one member that I am aware of is the Microbiotheria order, which contains a single species of primate like marsupial. Both of those could be contenders.
I would put forward the naked mole rat as a possible contender, it looks odd, though recognisably rodenty and has an usual social set up around a 'queen' rat, more akin to that seen in bees or ants.
The Aye Aye could win a few votes because it is a nocturnal lemur inhabiting the 'woodpecker' niche in Madagascar. It has a very long thin finger which it uses to hook out grubs in wood, having gnawed a hole to get beneath the bark.
I'd also put forward the Okapi which is a relative of the giraffe which looks like an animal which couldn't decide whether to be a giraffe, antelope or a zebra. It has stripey legs, big diamond shaped ears, a blue tongue, moderately long neck, lovely velvet brown coat.
Hard to argue with Jesus was a hippy's proposal that humans are the oddest overall.