Hi there,This sounds like a power supply issue and my best guess would be a bad/burnt connector or cold solder joints but lets try the easy stuff first.
There are 2 ribbon cables have you tried reseatting them?These ribbon cables do go bad due to heat coming from the boards.
Edit-I was looking back at some of my notes and found more info for you,On the random reset problem, on almost every Whitestar system, the problem has always been the CPU board’s CN2 connector. This brings +5 and +12 volts from the power driver board below it to the CPU board.Here's a pic,Let me know if this worked or not I have more notes on file

board (resized).jpg

Ok,Sounds good

Ok,Let me ask you this just to rule it out while i look through more notes.
Is the game plugged into the power outlet by itself or with other things?
Are the boards in the backbox in good shape?Any signs of hacks/burns/repairs ect.?
Any signs of battery corrosion?anywhere in the area where the 3 AA batteries are at.

Ok,The Whitestar board system is a very reliable system the only other thing that would cause resets besides the connector is in good/not burnt shape would be
replacing BR21(bridge rectifier).But here's the ass kicker about BR21 sometimes testing this bridge with a multimeter will test good/proper working order,Changing/Replacing BR21 would be the next best move.

Here's a few videos of the Whitestar system,Keep in mind Sega and Stern in many of their games use the same exact Whitestar system.Watch the videos and see if it relates to some of the problems your having(Videos made/provided by Clay..Thank You Clay).