"Matrix" is on his computer screen earlier.
Since Neo is seeking out what is the feeling ('like a splinter in his mind') that's bothering him, he may've realized that this meant to be interpreted as the answer to his question.
Furthermore, it's likely that Neo actually had some of the events from the 1996 script draft as an untold backstory. In that, the "Matrix has you hack" occours inside a chat/irc session where they're discussing the Matrix and some connection to Morpheus. The scene seems to be shortened to keep up with the pace of the movie, but it's still possible that Neo took part in conversations like that before. (Sorry for the long quote.)
A computer screen.
We are on-line, inside a chat room called "The Matrix."
It is an exklusive web-site where hackers hang out.
SCREEN
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
game.
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!!!
We drift back from the electric conversation entering --
INT. NEO'S APARTMENT
[...]
NEO, a younger man who knows more about living inside a
computer than living outside one.
NEO
Fuckin' idiots don't know shit.
He finishes his cereal and is about to disconnect when an
anonynous message slices onto the screen.
SCREEN
Do you want to know what the
Matrix is, Neo?
Neo is frozen when he reads his name.
SCREEN
SUPERASTIC: Who said that?
JACKON: Who's Neo?
GIBSON: This is a private board.
If you want to know, follow the
white rabbit.
NEO
What the hell...
SCREEN
TIMAXE: Someone is hacking the
hackers!
FOS4: It's Morpheus!!!!!
JACKON: Identify yourself.
Knock, knock, Neo.
A chill runs down his spine and when someone KNOCKS on
his door he almost jumps out of his chair.
(A funny thing is that it seems that both Neo and Trinity are not actually logged in but talking via hacking.)
(edited)
The hackers were already discussing these topics - it was so hot that the chat room itself is called "The Matrix". It's likely that it was like an urban legend, connected to Morhpheus. He was possibly spreading the word on boards/via spams/in chat rooms. Neo's comments imply that he already knows and is far more serious about these. He did a research on the topic.
Neo has this feeling that bothers him that "something is not right with the world". He tends not to follow rules that he feels a limit put on him. The strangeness of this feeling he can't articulate that he feels that these limits are put on him on purpose. He naturally question anything that he feels connected to this feeling. This includes authority and control like government decisions, false communication and media influenced social control. (Morpheus further extends the connected topics in Hotel Lafayette in the draft.)
Neo could start from his opposition and then learn about Morpheus whom is talking about this very thing everywhere. He gets interested in him and learns as much as he can. Neo must be quite into this topic since he considers the hacker's conversation "noobish" Fuckin' idiots don't know shit.
Alternatively, it could be that they know Morpheus first because he is known as a famous terrorist. Neo is a hacker, he is likely to consider/explore alternatives to official explanations to terrorist attacks (or simply view most news as cover stories right from the start).
It's also possible that simply it's known that Trinity is connected to Morpheus. Neo knows [s]he (The Trinity). It would be natural from Neo to do a search on Morpheus. Morpheus is also legendary, what would be enough on its own merit.
Either he just started to look for similar thinkers, or heard about the Matrix first then Morpheus, or vice-versa, he started to study the latter two. Morpheus is trying to free minds, while also looking for the One, so it's likely that he tried to hint as much information about the Matrix as it's humanly possible.