Do a Prime95 Large In-Place FFTs (setting in Prime95) stress test to make sure CPU doesn't overheat, power circuit is stable (no bluescreens or resets).
Do similar with Furmark for the GPU or IGP, or GPUTool:
These should both be allowed to run for several minutes at a minimum, the longer the better but I could see staying in someone's home or at a meeting place for long, being excessive (see argument made at the end for a 2 day warranty).
If the systems will be used for 3D gaming, you might want to use a more elaborate gaming benchmark like 3DMark (or whatever the cool kids use for that these days
Another thing to do is put up a solid white, then solid black image, to look for screen deformities or dead pixels.
Bring a flashlight and look into the fan intake and exhaust areas. If they are clogged with dust the system has been heat stressed. If they are clogged and there is no access panel, some surgery may be needed to disassemble the laptop to get to the dust to clean it out. "Some" even need the entire mainboard pulled out just to clean dust. Also observe whether the fan is making a racket (sign of wearing out) or spins smoothly.
Research the make/model spec'd runtime on battery power. Note where the battery % is when you start using it, and where it is after you stop. Doing the stress tests you will drain the battery faster than the manufacturer spec since it's for "best case" but you don't want the battery to be tanked in a very short period of time... but of course, on an old laptop you might be assuming you need a new battery anyway, but it would be nice if the one that comes with it still holds a decent charge in cases where they forget to recharge it and need it to work right away for class or whatever, have a viable 2nd battery.
Don't force Linux on them (lol) they need to have maximum usability and peer support for it.
You could use one of various HDD smart error detecting apps if you intend to reuse the existing HDD, but frankly I would pop an SSD in if they don't already have SSDs.
Have a widget to plug into any of the external sockets (USB, headphone jack, etc) to check whether they all work, unless that socket is unimportant (for example HDMI IF it would never be used with an external monitor).
Take the battery out and (gently, while doing one of the stress tests for higher power consumption) wiggle the power cord to see if the power socket is damaged. This is a very common failure point. Some have a separate power board the socket is on, while others can be fairly complex (or at least time consuming) to repair a socket soldered to the mainboard.
Wiggle the screen to ensure it has no hinge area cracks.
If the seller is open to it, you could draft up a purchase agreement that includes a 2 day warranty to allow you more time to investigate. You can argue that this allows you to skip the more lengthy tests so they don't have to entertain that for so long.