Android asks me for USB debugging permission on one computer, but not another
This was due to missing udev rules on my laptop which I'd forgotten I'd set up on my tower. You can find information on setting up udev rules elsewhere, but I'll post the exact steps I took here (which I got from my company's internal help wiki - remember we use weird hardware, so I can't guarantee this will work in all cases).

Create a file at /etc/udev/rules.d/51-android.rules with the following contents:

SUBSYSTEM=="usb", ATTRS{idVendor}=="0bb4", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0e79", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0502", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0b05", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="413c", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0489", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="091e", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="18d1", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0bb4", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="12d1", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="24e3", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="2116", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0482", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="17ef", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="1004", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="22b8", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0409", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="2080", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0955", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="2257", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="10a9", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="1d4d", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0471", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="04da", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="05c6", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="1f53", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="04e8", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="04dd", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0fce", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="0930", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="19d2", MODE="0666"
SUBSYSTEM=="usb", ATTRS{idVendor}=="1bbb", MODE="0666"


Then run:

sudo chmod 644 /etc/udev/rules.d/51-android.rules
sudo chown root. /etc/udev/rules.d/51-android.rules
sudo service udev restart
sudo killall adb


Disconnect and reconnect the device physically, then start using adb.