I've dealt with a few of them over the last few years, it can be very entertaining.
The way to do it is to set up a virtual machine with a
packet sniffer on it. If you use all the old tricks that you would have learned on the helpdesk, you can even keep them on the phone while you set it up.
Here's a few of the classics to get you started;
"Yes...Ok...right...ok...right..yes...<15 minutes later>...no, sorry, I'll never remember all this. The computers in the other room: If I go in there, would you show me how to do it?...."
"it's a terrible slow old thing, let me start it up <ten minutes later...talking about the grandkids, especially the oh-so-clever favourite who built the computer for you is an excellent way to pass the time> oh, looks like it's frozen, I'll just turn it off and start again..."
"Start button? Oh, he must mean the button on the front of the hard drive! OK...it's shutting down now...OK, it's off, now what?"<20 minutes, easy>
Once you've got it all set up, let them talk you through downloading and running the LogMeIn software on the virtual machine. don't make it too easy for them, now...Did you know the app that you download from LogMeIn is only good for five minutes? If you haven't got it running by then, you'll need to download a new one and try again! Hours of fun and excitement for you both!
So, let them connect and then use the packet sniffer to identify the IP address their connection is coming from. Also, here's a fun tip - the local session takes priority over the remote session, so if you are moving the mouse, even just a little bit, they can't! Fun!
Once you've had your fun, and you have their IP address, you can let the
authorities know, and their internet connection will become a very interesting place. Briefly.