"For the newcomers, OBC stands for On Board Computer."
And as everybody who owns an E31 knows, some of the pixels in the OBC's display eventually go dead.
The cause is a bad connection inside the OBC that permits dirt and moisture to creep in.
In theory, it is possible to remove the OBC and then clean the connection.
The more expensive fix is to replace the entire OBC, which sells for about $250.
If you are mechanically adept, you can remove and replace the OBC yourself.
1) Remove the vent grill directly above the console. There are two upper and two lower tabs that need to be depressed in removing the grill. Make sure to slide the unit to the right as you pull up because the grill piece is locked behind its neighboring part. Put the vent on top of the dash. (I put a cloth down on the dash first, the air vents are a little dusty.)
2) Slide a clean putty knife between the bottom of the OBC and the dash face plate. This ensures that the catch mechanism for the OBC is depressed. The catch mechanism on the bottom almost runs the full length of the unit.
3) Underneath the removed vent you will find a sheet of black plastic held in place by a small Phillips screw on the left side of the opening. You don't need to remove this sheeting. You can maneuver your hand around the plastic sheeting on either side to gain access to the back of the OBC unit.
4) Gently push the OBC out on both side till you fell the unit stopped against it's latching mechanisms. Using another putty knife, or a small flathead screw driver, depress the latch mechanisms. There are three on the top side evenly placed along the top. The unit should easily slide out at this point.
5) If you are just replacing the unit, simply unlatch, reattach and slide the new unit into place with a click.
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The OBC is probably one of the most notoriously failing electronic items on BMWs.
It's a rather stupid construction, as it relies solely on a tiny plastic piece to hold a 200-or-so-pole connector in place. We´re lucky not to be in outer space with a ship of that build quality.
I eventually got around to fixing my own OBC. I must warn you, this is one serious "can of worms", they really built the OBC to be unserviceable - except for changing the bulbs, which can be done from the outside. I haven't figured out the "definitive DIY fix" yet, but there is a simple method, the "QD Solution" to the pixel flicker.
Firstly, you need to determine whether you have a "Siemens" or a "Borg"
OBC. The factory part # sticker will tell you this. The Siemens is basically self-explanatory once you figure out how it's opened, the Borg is a real PITA (just like its Star Trek counterpart 
and needs extra caution when opening.
The LCD is connected via a flexible lead (etched ribbon strip). One end is sealed into the LCD, the other is glued onto the printed circuit board (PCB) and is held in place by means of a rubber-like padding under the plastic lamp reflector assembly. The problem area is, of course, where the flexible cable meets the PCB. Dirt "creeps" in via the air travelling through the front panel (you can see the black stuff inside the white plastic), and eventually gets under the ribbon-cable. Smoke is of course detrimental to a setup like this.
To clean it properly, one wants to remove the ribbon strip, but after that you really have some trouble putting it back in place as the glue is a one-time-only type, and there is some sort of heat-molded graphite stuff connecting each of the poles. I still haven't figured out how to fix this, so don't try to remove the cable.
What you do instead, is to spray contact cleaner from the back, so that the fluid goes between the PCB and the ribbon strip. I even tried spraying directly from the front, in the lower part of the LCD, and this actually worked on my friend's -95 328iA cabrio (also equipped with a "Borg" OBC). So, that should be tried first before the procedure below.
To clean "Borg" brand OBC from inside (the Siemens is similar but needs no further explanation to open):
1. Ignition off. Lights off. Take out the OBC by first removing your car stereo (disconnecting it is not necessary if it has original cabling) then pressing the plastic flap under the OBC inwards. The plastic flap covers 2/3 of the OBC bottom width and has two hinges at each end. Be careful if you have the wooden center console, the small strip of wood between the stereo and the OBC cracks easily. The OBC pops out. Remove the connector (one on Borg, but I've seen Siemens models with two connectors). Connector comes off by pressing the striped plastic "button" and folding the plastic lever over it. It's a rather crappy plastic connector, so be careful.
2. Make sure you have a reasonably anti-static work surface and clothes.
3. Disassemble the OBC: press the four black plastic hinges inwards, loosening the corners one at a time. The hinges are located at top and bottom rear corners. The back piece will now come off.
4. Locate the two black "fork type" plastic hinges holding the PCB in place, they are equidistant from the center of the PCB. (I think that "PCB" stands for printed circuit board. PMB) The plastic hinges are pressed together using a small plier or similar, allowing them to slip through the rectangular hole in the PCB. There are two white plastic hinges on the outer sides of the OBC holding the whole lamp assembly, these need to be depressed a little to allow the PCBs and LCD to come out of the box. It takes a little fiddling and some force to get it all out, but it can be done. Be careful not to apply too much pressure on the LCD itself, as it is made of glass. The LCD holder is a frame that runs all around the LCD, this has a tendency to get stuck on the way out. More fiddling required, and a good level of tolerance.
5. Now the white connection strip between the backlight lamp assembly and the LCD should be visible.
Do NOT attempt to remove the lamp assembly, it is soldered and holds the flexible cable in place. Clean the connection pads on the flexible cable with spray-on cleaner. Suitable cleaning agent is an alcohol or freon based spray, or any good quality contact cleaning spray without oil. Do NOT use 5-56 or similar greasy products, or graphite spray. Do NOT attempt to loose the flexible strip from the PCB edge connector (it won't go back that easily, as I've found out...).
7. Clean the LCD front while you're at it (you may find collected grime at the edges).
8. Reassemble in reverse order. Make sure the flexible cabling is not damaged, and that the LCD is in no way bent.
