I would double check anything and everything to do with that GI relay first (relay, soldering, connectors):
No Backbox General Illumination.
If the backbox General Illumination is missing, of course first check the A12 transformer panel fuse F8. If this fuse is Ok, then check the backbox GI (A) relay. The "A" relay has a single normally closed switch which completes the power path to the backbox GI. Sometimes the A relay switch can get mis-adjusted and cause the backbox GI to never work.
http://www.pinrepair.com/sys3/#gi

I seem to recall that I had one or two bad GI bulbs causing all them to be out in the backbox when mine was out. If that is the case, then they are all in series. Then again, might had been adjustment to the 'A' relay that did it.

Socket problems are also a common problem, so yes - one defective bulb can do,it as well

Unfortunately it appears that Gottlieb was using organic flux when they were building the system 3 machines. The problem with this flux is it can be conductive (this info came from an ex-Gottlieb designer). For example when Stargate came out there were reports that 1/3 of all machines that out of the box had some kind of light problem. After locating the socket that was causing the problem it could be easily moved around and cracking type of noise heard, indicating a short..

Well over 2 years ago when I fixed my Freddy backbox GI light issue and I just happened to see a couple of bulbs that turned black but I happened to had messed with the relay adjustment the same day too. Luckily, my Stargate hadn't had any GI issues since I had it for about 7 years now.

The contacts on that A GI relay are highly suspect. Put a jump wire on it for test (short across the terminals for the backbox lighting) and see if they come on. If so, the contacts need to be cleaned or adjusted. I had a Freddy do that, the contacts were misaligned but appeared to be touching. It was intermittent until I worked no the contacts. Now Freddy has backbox lights. I am afraid of the dark.

For what it is worth - I just had the same problem - half the backbox GI in my Freddy was very dim or barely working, and since half - I knew the fuse or relay couldn't be the problem...simply tightened the very loose ring terminal (which was right after the last bright bulb in the string) and back to normal!