According to the story Lilith was made out of the same clay as Adam and was equal to him, but later turned evil by killing her own children and God expelled her on the desert where she became a demon who's raping people and eating children and then God made Eve out of Adam's rib, at least for me something smells bad in this story. Hawa in Hebrew means source of life if I remember correctly, so originally it could be just a title rather than name.

In Canaanite mythology Adammu who originally was a divine being and later became mortal received from gods a good natured woman as a wife who also was divine and they were equal, authors of the book "Adam, Eve and The Devil: New Beginning" associated this woman with goddess Kubaba, but I think that it could be Lilith-like figure as well.

It's not impossible that Yahwists who relegated status of women removed figure who's symbol of empowerment of women and introduced new one who has lower status than a man.