I've booted LMDE 7 off of a USB stick, found an Ethernet cable, found the spare trackball (Logitech M575S - great design in every way except that the buttons are terribly mushy), and ...
It can play a The Wide World of Jank (thank you, @BigBadBiologist) at 720p and 60 fps. Does so at 20% processor load, which doesn't seem terrible.
That's promising enough that I can continue with this one. I'll find the wireless card, replace the mechanical drive with an SSD, and find it a home.
I'm not going to spend too much more time on this tonight, but I did pull one case fan.*
This is what the grill looks like beneath the fan. Honestly, I haven't seen stuff like this since tech support in the '90s, when smoking was more common.
* I'm going to use the fan to diagnose a boot error message in another PC from yet another relative. I need better hobbies.
Would have done the same thing. My old PC was in the room with the router. My work laptop docking station is 5 or 6 ft from the router. I just ran a cable and called it a day.
I pass around older iOS devices in a similar manner. If usable, spread it around!