Deep-web searches like pipl and peekyou will find social networking sites, and from there you can monitor activity to see what they're up to, who they're talking to etc. Another possibility would be to make an account on something like Facebook under a false name and add a bunch of the person's friends, maybe say you're a friend of a friend or some obscure friend's sibling or something, and then you can see a little more because you fall into Facebook's "friends of friends" category rather than just being shown the "visible to everyone" stuff. The person in question wouldn't know because you haven't added them as a friend, just their friends, so there won't be any suspicious names in their friends list.
Depending on how tech-savvy your stalker is, there are also more advanced methods that involve snooping on their internet connection. If the target uses a wireless internet connection, that connection can be broken into (how easy this is depends on how well they've set up the security; could take anything from minutes to weeks) and your stalker can watch more or less everything the target does online with what's known as a man-in-the-middle attack. Basically, the stalker tricks the target's computer into sending everything to the stalker's computer, instead of to the wireless access point they'd normally use. The stalker then sets his computer to pass it on to the wireless access point so that neither the access point nor the target notice, and vice versa, everything from the access point goes to the stalker, who passes it on to the target. On the way through, though, everything is copied onto the stalker's computer, so the stalker can decode it and read through to see everything the target is doing online. This kind of attack usually goes after logins and bank details, but it can be used to find anything in the target's internet usage that interests you.
If the target is on a wired connection, it's somewhat harder, but it's still possible to grab their internet traffic - it just involves messing about with any switches and routers they're connected to so that their traffic gets forwarded to somewhere that the stalker can see it and take copies.