I found an "Overview of Lilith" from an Alan Humm. His (own) biography page lists him as a PhD in Religious Studies but a quick searching of the internet didn't reveal much information about him.
The relevant parts of the Lilith notes (emphasis added):
Lilith is the most important of a small collection of named female demons in Jewish legend. Historically, she is actually older than Judaism (at least Judaism as defined as a post-restoration phenomenon). Her earliest appearance is probably in ancient Sumer. Although it is far from certain, she may be a minor character in a prologue to the Epic of Gilgamesh. In the ancient world she also sometimes appears in magical texts, amulets, etc., intended to thwart her activities. She appears once in the Bible (Isaiah), in a context that associates her with demons of the desert, and again in some Dead Sea Scroll passages clearly based on the Isaiah reference.
And later in the article (emphasis added):
Somewhere between the eighth and tenth centuries, CE, she makes an appearance in a satirical work entitled the Alphabet of Ben Sira. It is here that she is first given what has become her most famous persona: the first wife of Adam (before Eve). In this story, she is created at more or less the same time as Adam, and, as was Adam, out of the ground. Because of this she tries to assert her equality -- an assertion which Adam rejects. [...]
While it is true that there was a rabbinic tradition that Adam briefly had another wife before the creation of Eve (Genesis Rabbah), there is a great deal of doubt as to whether Lilith had any connection at all to this first wife of Adam story prior the publication of the Alphabet. The satirical nature of the Alphabet casts further doubt on the authenticity of this Lilith connection. [...] As a midrash, it also helps to solve a problem that arises from the fact that Genesis 1 has mankind created "male and female," but when we get to Genesis 2, Adam seems to be alone and in need of a partner.
There are other interesting details about Lilith (associations with feminism, vampires, baby-stealing) but the gist appears to be that the origins of Lilith as "Adam's wife" are directly traceable to a satirical piece that was clearly never intended to be part of the Holy Bible. The concept of Adam's pre-Eve wife does have some merit in the Genesis Rabbah but the stories are drastically different. From the same author as above (emphasis added):
Genesis Rabbah contains two references to a tradition that may be related to the Alphabet of Ben Sira, although probably not directly to Lilith. Here we find a much earlier example of the idea that Eve was Adam's second wife. But in this tradition the first woman is unnamed, and there is no reason to think that she should be identified with Lilith. In addition, the reason given for the failure of the first marriage is entirely different than that given in the Alphabet. Whereas Ben Sira has the problem center around sexuality and dominance, R. Judah b. Rabbi has Adam apparently disgusted by seeing the process of the woman being created "full of discharge and blood" (although this may instead be a reference to menses).
Based on this information, Lilith was never "meant" to be in the Bible and, therefore, could not have been written out. Even if the concept of Adam's pre-Eve wife were somehow excluded, that wife would not likely have been Lilith.