Depends on your code-foo and, frankly, how evil you're willing to be.
If the person is a solo operator in it for cash and profit, chances are they've left a trail online and in brick/mortar drop locations. If it's a centralized criminal operation, this is not the sort of thing you want to undertake extra-legally - better just to have your friend file a credit freeze and diligently review his/her credit report a couple times a year.
If this is just some dork jerking around with phony facebook entries, that is typically someone you know pursuing some sort of emotionally-motivated revenge campaign. I recommend you ignore it and move on with your life. If the crimes are something the police are already interested in, they could easily contact the ISP associated with the IP and request an ID. Unfortunately, that only works as far back on the trunk as the first server stop, and the cops have to be interested.
If you end up with a name or address, the police can handle it. Or, you can PM me for some more tips.
The dark side, this is...