There is a kind of endless and indirect conversation that takes place among TV broadcasters and writers, all of whom listen to each other's shows as a matter of course. I think the name "nine-eleven" gained its dominance as part of this conversation. It does not follow the usual format for naming dates of importance, and I suspect it could not have become the predominant name if people had not heard it spoken aloud. "9/11" might have been read "September Eleventh" in familiar practice if people had only seen it in print, but plainly there was a sense somewhere that a different sort of name was appropriate and I suspect it was able to spread by sound.
Early in the days after the September 11th attacks, other names were also to be heard. Some people said "September Eleventh", which is the traditional way of naming dates of importance. "September the Eleventh", a little more formal, or reminiscent of British usage, also cropped up. Some people also said "nine-one-one", which suggested one of the numerological explanations for the terrorists' choice of date. But a year after the event it is clear that "nine-eleven" has become the name I hear most often. I have been quite conscious of the trend, because this is not a name that sounds comfortable to me.