Probably the service indicator board. You might need to replace the batteries or possibly the whole board if the batteries leaked and ruined the board. Go to the
Unofficial BMW Home Page, look under Resources Unique to this site and select The Unofficial E30 Home Page. Once there towards the bottom of the page you'll find a section on Electrical stuff. Select the Service Indication Lights FAQ and look for a section on replacing the batteries on the SI board. In there he says you might want to buy a battery holder with wires coming out of it and solder the wires to the board so you don't have to remove the whole instrument cluster the next time it happens. Although it really shouldn't happen again in a long time. It just might be easier than trying to re-install the batteries on the board.
If the batteries aren't the problem it is probably the board itself. I think the dealer charges a lot for them ($130+) so you might want to find another source. fporro said he had some in a previous post so you might IM him. But pull the board first and check the batteries. That may be all you need to do.
Hope that helps.
Steve