Check all segments on self test (Press and hold "autohold" for 87-V, Press and hold "hold" for 87 and 87-III)
Select Resistance function
Check lead resistance should be 0.something, lower is better. (not a dealbreaker, but make sure there's no flashing OL or anything.)
Place red probe in ma jack ("wrong way") resistance should be about 1K. "OL" means fuse is blown.
Place red probe in A jack - should be low resistance. (0. something)
Check backlight
Bring a 9v battery and a 1.5v cell for a quick sanity check.*
(the next test only applies to 87 and 87-III)
Press and hold range switch at turn on for segment check - should get
0 - AC volts
32 - DC volts
64 - DC millivolts
96 - Resistance
128 - diode
160 - ma
192 microamperes
That's what I can think of off the top of my head. I'm sure the meter gurus will respond shortly.
* In most cases, if the reading is reasonable, the meter is fine. If it's been damaged, the readings will be WAY off. I suppose there's the very rare case where it could have drifted 2-6 percent, but I've never seen it. Every Fluke meter I've seen or worked on was either in spec (or very close for older meters) or was very obviously wrong. That is, you don't need a precision voltage source to check a meter in most cases.