In terms of the theatrical movie, it's not really apparent what Neo does (or doesn't) know about The Matrix at the start of the film. Certainly he's been trying to get into contact with Morpheus, a known cyber-criminal, for reasons that aren't really clear and he seems to have some idea that the world isn't right, but can't find any way to vocalise it. He has what Morpheus refers to as a 'splinter in his mind'.
Trinity tells him that...
The Matrix has you...
...shortly before Choi and DuJour turn up at his apartment and take him to the nightclub, so we know he's heard about the Matrix at least once before his conversation with her.
In earlier versions of the script, Neo is watching other hackers discussing rumours about something called the Matrix. None of them have a clue what it is, but they do know it's something shady.
We are on-line, inside a chat room called "The Matrix."
It is an exklusive web-site where hackers hang out.
JACKON: I heard Morpheus has been
on this board.
SUPERASTIC: Morpheus doesn't even
exist and the Matrix is nothing
but an advertising gimmick 4 a new
TIMAXE: All I want to know is
Trinity really a girl?
LODIII: 87% of all women on line
are really men.
QUARK: The Matrix is a euphemism
for the government.
SUPERASTIC: No, The Matrix is the
system controlling our lives.
TIMAXE: You mean MTV.
SUPERASTIC: I mean Sega.
FOS4: ALL HAIL SEGA!!!