What does it mean when the battery voltage light stays on in a 1988 Honda Accord?

It could mean a few things. There's a voltage range in which that light stays off, which seems to be between 11.5v and 14.5v. The light comes on when you're either above or below that range. Start by checking the alternator fuse. If that's good, and it almost always is, do a voltage test. Too low and the alternator is probably bad; too high and the voltage regulator went out. (Which is basically the same thing, since the VR is inside the alternator case.) Check to see if alternator is charging battery. With volt meter connect to battery and vehicle not running check voltage - start vehicle and observe volt meter should be around 14.2 volts - if no different running or not or if lower with vehicle running alternator is not charging battery and should be replaced. If alternator functioning properly it is probably just the "trigger" wire on alternator.