Very little is known from deities of the period, so this one is no exception. But I wonder if these messages carved on stone have survived longer than what our UTF-8 encoded text files may ever hope to.
The Dictionary of Deities and Demons in the Bible has an interesting article on Lilith, but it doesn't provide substantial evidence that this creature is the same as represented by the Babylonian carving. Lilith is attested to, most likely, in other Mesopotamian literature, but distorted by later Jewish and Christian lore.
This is one of the reasons why I find ancient Near Eastern materials so fascinating. We're always discovering something long forgotten, and then it provides evidence roping it back into other surviving literature, stories, or mythologies (the Epic of Gilgamesh being one such work).