If this story is widely accepted in the Jewish tradition and religion then it HAS to be part of the Christian tradition considering it piggybacked off of the Jewish religion.
You're making the assumption that Judaism's myths and traditions became frozen in place when Christianity developed. Judaism has continued to develop as its own religion for the past 2000 years.
^^^AND it reinforces the lack of knowledge of the history of their religion that we mentioned earlier.
Without the precursor of Judaism, there IS no Christianity. It amazes me how many people don't understand that simple fact - even when I was still a believer (and thus comparatively young), I knew that much. Now I find myself wondering how I came by that piece of knowledge, because it's certainly not common.
We don't have to look far to find that history, either: screwtape kindly linked an excellent Wiki article.
People in this thread have mentioned that Lilith is mentioned in the Talmud. Yes, she is, specifically in the Gemara, which was published a couple centuries after
Christianity's split from Judaism.
Per the wiki article, there is only one pre-Christian Jewish reference of Lilith in the Dead Sea Scrolls:And I, the Instructor, proclaim His glorious splendour so as to frighten and to te[rrify] all the spirits of the destroying angels, spirits of the bastards, demons, Lilith, howlers, and [desert dwellers…] and those which fall upon men without warning to lead them astray
She appears in the context of a list of demons/spirits/etc. There is no other description of her. (That's the same depiction in the Talmud, incidentally.)
Lilith as the first "Eve" does not appear until the 8th to 10th centuries CE, hundreds of years post-Christianity. This was further developed throughout the Middle Ages in Jewish Mysticism.
Lastly, the article depicts Lilith's legend as a folk legend appearing in Jewish Mysticism, but not ever really being a central part of Judaism. It concludes noting that Lilith appears in artwork, the occult, Wicca, Satanism, and mystery traditions... but makes no mention of her as a tenet of modern Judaism.
Thus, the Lilith legend evolved almost completely outside of Christianity, and has nothing to do with the origins of Christian scripture.