When I was a teenager I'd do stupid crap and you can sometimes gain info from an IP.
Back then I had a tool that would resolve an IP to an ICQ uin, from that I'd look at the profile for first/last name. If not, there would often be an email or zip code or something I could work with. Google stores tons of crap even from message forums you can learn tons about someone (that left their email). Can find out their likes/dislikes and other things and put stuff together.
Of course you could run nmap on them and try to find anything open. Not sure if this is legal where you are, but I don't see why not.
Check for open shares or something. This is a grey area so I won't condone it nor give you help there
Maybe they run httpd, try IPs above and below theirs in the same area. Perhaps they have more then one IP and run multiple comps off their connection. If they run a website, go there. Find more info. Google for answers.
Check who owns the IP block. You can often level it down to an area with it. You can find out the ISP and which area that ISP serves. If it is someone in his/her city and they know the person they could think if they know anyone who uses that ISP.
Google the IP address itself. If its static there may be some results involving it. Who knows, another forum post where they publicly say who left the message "IP x.x.x.x at 12:00 EST" etc. You can find many things if you are resourceful.
Way back when, three years ago, I used to sell online game items. Someone ripped me off using a stolen CC. The only information I had was their nickname. Short story, within 72 hours I knew his first/last name, which school he went to, when his parents divorced and stuff like that through a little creative ingenuity of my own. Its a grey area but I did it all legally. And ended up getting my couple hundred dollars back
It only took me an hour each day, so probably 3 hours total. Plus it was just for the hell of it to call him up and hear how nervous he was.