"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.
(edited. Sorry, grew a bit long, hope it explains or at least makes more sense :D )
Neo has this feeling that bothers him that "something is not right with the world". He tends not to follow rules that he feels as limits put on him. The strangeness of this feeling he can't articulate that he feels that these limits are put on him on purpose, one that is intentionally hidden from him. The limits won't be bothering him if there were reasons for them, but it's likely from his attitude that all he got for his questions were menace, threats and oppression (or been ridiculed).
So he naturally questions anything that he feels connected to this feeling. This includes questioning authority and control like government decisions, thinking of false communication and media influenced social control. But just cannot getting an answer to his question of what is bothering him from the shadows and why it is hidden, he keeps exploring the previous topics what leads him to Morpheus (see below).
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.
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!!!
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.
Fuckin' idiots don't know shit.
He finishes his cereal and is about to disconnect when an
anonynous message slices onto the screen.
Do you want to know what the
Matrix is, Neo?
Neo is frozen when he reads his name.
SUPERASTIC: Who said that?
JACKON: Who's Neo?
GIBSON: This is a private board.
If you want to know, follow the
What the hell...
TIMAXE: Someone is hacking the
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.)
The hackers were already discussing these topics - it was so hot that the chat room itself is called "The Matrix". It seems that the Matrix and Morpheus were connected to some new game but some still believed in them (much like the Publius Enigma). It's likely that it was like an urban legend (about the hidden control, of course), connected to Morpheus. Neo's comments imply that he already knows and is far more serious about these.
Neo could drove from his opposition to authority and then learn about Morpheus whom he feels is talking everywhere about the very thing that bothers him. He was possibly spreading his word on boards/via spams/in chat rooms. Neo, hoping that he may be real and (possibly from Morpheus' themes and style) that he can finally find an answer, does a research on the topic. He learns as much as he can, and must be quite into and serious about this topic since he considers the hacker's conversation
Fuckin' idiots don't know shit.
Alternatively, it could be that he knows Morpheus first because he is known as a terrorist (the movie allows him to be even famous while this latter draft excerpt would imply that Neo was hacking his way to the information on him). Neo as a hacker, is likely to consider/explore alternatives to official explanations to terrorist attacks (or simply view most news as cover stories right from the start) and thus get familiarized with Morpheus.
It's also possible that simply it's known that Trinity is connected to Morpheus and Neo knows about "her/him" (The Trinity). It would be natural from Neo to do a search on Morpheus. Morpheus is also called "legendary" in the draft, which 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.