I've been thinking about this. They've given very little info on Babylon.
Unique Unit would likely be a Bowman because they probably aren't going to think of anything better (and because Archers were extremely important in mesopotamian warfare). I can't think of a second UU (well, aside from pointing out the Chariot/Horse Archer warfare of Mesopotamia, but nothing specifically Babylonian), so a UB could be possible. But the only thing I can think of would be a Ziggaurat and they were far more Sumerian than Babylonian (unless we want to preclude the possibility of Sumer later).
Now for the unique ability. The thing I think of with Babylon is it's enormous cultural and religious influence on mesopotamia, especially with religion.
That being said, the thing most people think of is the Code of Hammurabi. While I'm just ripping off the idea of the Code of Justinian in the expansion ideas thread, I'll suggest it'll be something related to that. Perhaps a courthouse bonus mixed with social policy bonuses.
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