The claim that Lilith was Adam's first wife, made from the same earth, who refused to lie beneath him and left Eden speaking the divine name, is repeated as though it were a suppressed passage of Genesis. It comes from a single medieval text, and that text is probably a satire.
Genesis contains two creation accounts. In the first, God creates humankind male and female. In the second, God forms the man and later the woman from his side. Medieval commentators noticed the discrepancy, and the story of a first wife is one way of resolving it. It is a reading of the tension, not a report of a lost verse.
The narrative appears in the Alphabet of Ben Sira, an anonymous Hebrew work of roughly the eighth to tenth century, most likely from the Islamic east. The work is a collection of stories built around proverbs, and it is bawdy, irreverent and full of material that reads as deliberately outrageous, it has been argued to be a parody of rabbinic literature, possibly aimed at rabbinic exegetical method itself, and possibly an anti-Christian or anti-rabbinic squib. Scholars disagree about its purpose; almost none read it as a straight repository of ancient tradition.
Earlier Lilith material exists and is different. The Babylonian Talmud mentions a lilith with long hair and wings, in passing, as a demon of the night. The Hebrew Bible has the word once, in Isaiah 34:14, in a list of desert creatures inhabiting a ruin, where it may be a night bird or a demon and translators have never agreed.
The rest is later Kabbalah, which developed her considerably, and the twentieth century, which developed her again.