Seems bad usb ports are on the rise. If the phone says it's charging or in usb mode when the cord is not plugged in, that's not good. The usb cord doesn't fit very tight, so don't let that make you think it's the sign of a bad usb port.
At least for now it 'seems' to be the case, that it's the usb port that is the faulty part.
Make sure gps can connect as well as wifi (and wifi stays connected)
That's the only main ones that comes to my mind. There may be other issues but most are very minor and doesn't effect the phone (raised corner of the glass, noisy case/volume buttons)
Just use every function on the phone to make sure it works. Phone calls work through the ear speaker and the back speaker.
Check to make sure the digitizer works 100%. Easiest way to do that is dial *#1*#. It will bring up the service test screen. There will be a number of useful tests that you can run.
If possible check to see if the flash counter has been tripped. Hopefully if it has been they'll just tell you. If they say no or I don't know and you want to make sure, boot it into download mode using a usb cable plugged into a computer (I think you do it with the phone off, battery out, and plugged into the usb port. While holding both volume keys insert the battery. If I'm wrong, please anyone correct me) When in download mode, where it says custom binary, it ideally will say No.
Not a big deal as a usb jig will reset that. But it will let you know that the person has at least rooted it. There may be a reason they're trying to get rid of it. If they are hiding anything it might be enough to get them to talk about why. Or at the very least let you negotiate a discount.
Sent from my páhhōniē