I just fixed this problem myself with the help of an article.
http://www.boston-bmwcca.org/bimmer/2002/10/radio.asp
I bought my wife a factory head unit-compatible CD changer from Crutchfield (USA Spec SCD-B2i) for X'mas, thinking the display problem was an easy fix...a bulb I could get over the counter.
I found that fixing the problem meant getting another radio, or work on the unit myself.
As in the article, there are two tiny resistors on the PC board that are involved with the display illumination that need the ends resoldered (they appeared fine, but when I used the tip of a small knive to check if they were firmly attached, the ends of two resistors broke free). They are just underneath the cassette player, towards the front of the unit.
They are accessable once the bracket with front PC board where the display is mounted [2 screws into the cassette mechanism] and the cassette mechanism [2 screws into bottom posts]are removed. The article photo gives just enough visual information to locate them.
It worked fine. No need to replace the bulbs.
It's easy and free, but there is a place in Santa Monica (forgot name, 310-396-7781) that will sell you a refurbished unit with trade-in for $200 (installation $20) .
The CD changer was mounted with stock BMW CD changer kit (with bracket and cover). Bracket mounts to pre-existing hole in rear deck with a bolt (washer, spacer)and to the side with two nuts on pre-existing posts. Over $100, but is an easy, neat installation. Bracket needs to have a mounting hole on each side to be slightly moved to accept CD changer mounting holes, no big deal - if ordering BMW OEM unit, you shouldn't have this problem.
A little work to get to the pre-wired connections in the trunk (trim panel above battery needs to be pulled back). One connector is a three pin wrapped with fabric electrical tape. It has the brown wire at the side... NOT in the middle (this is for cell phone? and may short changer). The second connector is larger, with shrink-wrapped wiring.
The head unit connecter plug was tough getting off. There is a outer piece that slides up first(make sure to continue sliding until the plastic pins completely disappear under it) then connector can be pulled straight off. It's on a little tight.
Happy New Year and Good Luck!
Linear Mode
