I had the same, or very similar, problem on my 96 328i. I fixed it myself for about ten dollars and it's been working great since I did it two years ago.
What mine did was go on and off randomly. There really was no distinct "pattern." The awkward part was that when the climate control display went out, it would engage the air conditioner-Extremely annoying.
It's not reliable for me to say that my problem was the same as yours. However, it's more than likely. If you haven't checked out this site at least give it a look through. If it's something you haven't tried yet, give it a shot. I ended up having to buy the correct size capacitor off of Amazon. They only sold them in bags of 10 though, so I still have nine of them laying around. If you need one just shoot me an address and I'll send it snail mail, no charge.
Here's the link: