I've done some digging into this problem and found a Bally display repair guide that had this to say:
"One digit is out on one of the displays
If you have a single digit out, then it's probably the the level shifter and digit driver transistors for that digit."
He goes on to conveniently list and label those transistors for Bally games, but no such luck for us Williams folk.
This excerpt is from a Williams display troubleshooting guide:
If you are having display problems (and you have fixed the power supply board), the next course of action is to check the UDN6118 and UDN7180 chips on the Master display board. The UDN6118's control the strobe pulses, and the UDN7180 control each segment in the display. Usually the UDN7180 is the one that fails (and are unfortunately hard to get and somewhat expensive). There are as many as four UDN7180's and UDN6118 chips on the master display board.
I'm looking at the schematic for this thing and see that each display (2 x 16 digit displays) has its own pair of each chip, so that narrows it down a bit more. It further tells us that UDN7180 controls segments. the schematic labels the segments that each pin of the 2 chips control, and this is NOT digit specific. If you have the same segment out on all digits this is probably the problem, but that's not my issue. So I have narrowed it down to the 2 6118 chips that control Strobe pulses (U17 and U18.) The schematic labels the output of these chips as STB 1-16 (googling STB reveals its just short for strobe) we have 16 digits and 16 strobe signals. Its an assumption, but if those numbers correspond to digits left to right then it seems U18 is my problem.
I'm going to grab a few IC sockets from radioshack today, desolder the 6118 chips and swap them. In theory if the problem digit moves, then I have identified the bad chip. If its fixed, It was a solder joint on one of these chips. If nothing changes, I'm out $2 and back to the drawing board.