"Matrix" is on his computer monitor earlier.
![Tha Matrix has you...][1]
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][2] 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.<br/><br/> We are on-line, inside a chat room called "The Matrix." <br/> It is an exklusive web-site where hackers hang out.<br/><br/> SCREEN<br/> JACKON: I heard Morpheus has been<br/> on this board.<br/> SUPERASTIC: Morpheus doesn't even<br/> exist and the Matrix is nothing <br/> but an advertising gimmick 4 a new<br/> game.<br/> TIMAXE: All I want to know is<br/> Trinity really a girl?<br/> LODIII: 87% of all women on line<br/> are really men.<br/> QUARK: The Matrix is a euphemism <br/> for the government.<br/> SUPERASTIC: No, The Matrix is the<br/> system controlling our lives.<br/> TIMAXE: You mean MTV.<br/> SUPERASTIC: I mean Sega.<br/> FOS4: ALL HAIL SEGA!!!<br/><br/> We drift back from the electric conversation entering --<br/><br/><br/> INT. NEO'S APARTMENT<br/>[...]<br/> NEO, a younger man who knows more about living inside a <br/> computer than living outside one.<br/><br/> NEO<br/> Fuckin' idiots don't know shit.<br/><br/> He finishes his cereal and is about to disconnect when an <br/> anonynous message slices onto the screen.<br/><br/> SCREEN<br/> Do you want to know what the<br/> Matrix is, Neo?<br/><br/> Neo is frozen when he reads his name.<br/><br/> SCREEN<br/> SUPERASTIC: Who said that?<br/> JACKON: Who's Neo?<br/> GIBSON: This is a private board.<br/><br/> If you want to know, follow the<br/> white rabbit.<br/><br/> NEO<br/> What the hell...<br/><br/> SCREEN<br/> TIMAXE: Someone is hacking the<br/> hackers!<br/> FOS4: It's Morpheus!!!!!<br/> JACKON: Identify yourself.<br/><br/> Knock, knock, Neo.<br/><br/> A chill runs down his spine and when someone KNOCKS on<br/> 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.
Neo's comments imply that he already knows about some of this and did a research on the topic. Seems that hackers were already discussing these topics.
(*edited*)
They may know Morpheus 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). Also, Neo has this feeling that bothers him so he may question anything that he feels connected to that feeling, including government decisions and media influenced social control, so he could got interested in Morpheus because of that. He must be quite into this topic since he considers the above conversation <del>"noobish"</del> `Fuckin' idiots don't know shit`.
It's also possible that it's known that Trinity is connected to Morpheus. Neo knows [s]he because she is a famous hacker in hacker circles (*The* Trinity). It would be natural from Neo to do a search on Morpheus. Morpheus may also be a known hacker, what would be enough on its own merit.
Once he got to Morpheus, he possibly found information on the Matrix there. Morpheus is trying to free minds, while also looking for the One, so it's likely that he tried to push information about the Matrix as much as it's humanly possible.
[1]: https://i.sstatic.net/etifr.png
[2]: http://www.scifiscripts.com/scripts/matrix_96_draft.txt