I don't recall a Lilith from any of the Christian stories. I only know this because I've read The Mortal Instruments series. Why is Lilith, the first wife of Adam, not mentioned in the Bible?

 

From the Anchor Yale Bible Dictionary:

LILITH (DEITY) [Heb lı̂lı̂t (לִילִית)]. Lilith is the Hebrew form of Akk lilı̄tu (the feminine form of lilû), which was a species of lesser deities in Mesopotamia known for their diabolical activities (Farber RLA 7: 23; Porada RLA 7: 24–25; CAD s.v. lilû). Very little information has been found relating to the Akkadian and Babylonian view of these demons. Two sources of information previously used to define Lilith are both suspect. Kramer (1938: 5) translated ki-sikil-líl-lá-ke4 as “Lilith” in a Sumerian Gilgamesh fragment. The text relates an incident where this female being takes up lodging in a tree trunk which has a Zu-bird perched in the branches and a snake living among the roots. This text was used to interpret a sculpture of a woman with bird talons for feet as being a depiction of Lilith (Frankfort 1937: 130 fig 1, 134–35; Kraeling 1937: 18). From the beginning this interpretation was questioned (Opitz 1932: 330) so that after some debate neither the female in the story nor the figure are assumed to be Lilith (Ribichini 1978: 31–33; RLA 7:25). The 1st millennium Syrian incantation from Arshlan-Tash, often cited as a Lilith reference also has been shown not to refer to this demon (Gaster 1942: 44, 50; Torczyner 1947: 29).

Amulets exist to ward off lilı̄tu and the lilû family. Lilith was known to attack women in childbirth, a characteristic she may have adopted from Lamashtu (EncJud 11: 246–47; Farber 1989: 4, 103, 117, 142–43). The new infant was in more danger than the mother since Lilith could suck its blood, eat its marrow, and then consume its flesh (RLA 7:23). To protect oneself from this demon one could write out incantations, wear amulets, or invoke gods or other demons, especially the king of the lilû, Pazuzu (Lambert 1968: 42, 46).

Two Jewish sources may be used to determine the activities of Lilith. The first of these are the four references to Lilith in the Talmud (Nid. 24b; B. Bat. 73a; Šabb. 151b; ʿErub. 100b). From these texts it is clear that Lilith has the form of a woman with long hair and wings, who bears demonic offspring. There is a sexual aspect implied in the warning that men who sleep alone could be seized by her.

The other source of information on Lilith and her relatives is the series of Aramaic and Mandaic bowl inscriptions used to ward off demons. Most of these texts came from a Jewish community in Nippur, though they date from the 1st millennium C.E. (Montgomery 1913: 76–78, 117, 155–56, 209, 244, 259–60; Isbel 1975: 17, 44–45, 108, 120–21; Geller 1986: 108–9). These texts confirm Lilith’s malevolent activities and they show her lurking around human habitations waiting for a chance to seize people. One incantation bowl bears a drawing of the shackled Lilith bound in chains around her neck, arms, and legs; she has been stripped bare and her hair loosed (Montgomery 1913: 154–55 no. 8 lines 2–3, pl. 8).

When only Jewish sources for the deity were known, it was assumed her name was a variant on lylh and thus she was understood to be the “night hag” (RSV Isa 34:14; ERE 4: 598). Once her Mesopotamian origins became clear the name was associated with Sum líl, “wind” (Meissner 1925: 201; Patai 1967: 207; RLA 7: 23).

Lilith was taken up by Jewish midrashic and cabalistic traditions and there is a rich literature which includes visions of her as a benign being as well as a demon (Patai 1967: 217–45). Currently Lilith is an inspiration for much creative writing and reinterpretation (Koltuv 1986: 126–27).