Possibly the battery?

System1 games with no battery or bad battery will show some pretty wacky high scores and credit numbers. It's often confusing, making you think there's a problem when there isn't one.

No matter what style back-up battery you use, make sure the battery voltage is getting to the 5101 RAM chip. The 5101 is three chips to the right of the Game PROM, along the top left side of the CPU board. Put the red lead of your DMM on the upper right pin 22 of the 5101 RAM, and the black lead on ground. You should get 3 to 4.5 volts DC if the battery backup is doing its job.

Be Sure to Zero Out the Game's Audit Memory.
After the new battery or memory cap is installed and working, I would highly recommend zeroing out the game's audit memory. If you don't do this and the battery was removed or dead, the CPU memory could have some wacky data which may cause some strange game issues or weird high score.

To zero out the audit memory, turn the game on, and press the white diagnostic button inside the coin door. This will show "0" in the credit display, and the audit information in all the score displays. To zero out this memory position, press the push button mounted on the CPU board, and the score displays should all go to "000000". Press the coin door diagnostic button once to advance to the next audit number, and again zero this out using the CPU board push button. Repeat this for audits 0 to 10. Press the coin door diagnostic button again, and then exit the audits by either powering the game off, or by opening the slam switch (or closing a tilt switch). This process will clear the game's audit memory.

Coin Door Switches - a Common Switch Matrix Problem.
Because system1 games do not have a free-play setting, users often press the coin switches inside the coin door with their fingers to add credits. This is fine, but while doing this often the user will move the "lock-out wire" so it touches one of the coin switch blades. This will short that return/strobe switch matrix line to ground, making the game freak-out. Usually a game won't start, or there will be some other strange game behavior. The game may not even boot, or may act like it is always tilted or slam tilted.

My suggestion is to set the CPU board DIP switch as follows to make repairing the game easier. After you have the game working, set the switches as desired.

DIP 1-8=all off (one coin, one credit). Another common setting is to make switches 1=on (and sw 2-8=off) for 1 coin 9 credits.
DIP 9=on (three balls/game, off=5 balls/game).
DIP 10=on (match feature on).
DIP 11=on (replay instead of extra ball, off=extra ball).
DIP 12=on (tilt kills current ball only, off=tilt kills game).
DIP 13=on (show number of credits).
DIP 14=on (play a tune when game started).
DIP 15,16 not used.
DIP 17,18=on (maximum credits 15).
DIP 19=on (make coin chute 1 & 2 the same coin value, important to be "on" if using the 1 coin 9 credit option).
DIP 20=on (chimes/tones when scoring).
DIP 21=on (show high score to date).
DIP 22=on (award 3 credits when high score beat).
DIP 23=on (play a tune when money inserted).
DIP 24 not used.
Having the switches in these positions will make troubleshooting a bit easier and consistent from board to board.

The battery way seem to work. But the voltage may be low. What voltage do you get on the ram chip pin with the machine off.
Where to check is listed in my first post.

Also the power supply capacitor C2 (220pf) limits high frequency noise on the +5VDC
            supply. If this capacitor opens, the bookkeeping functions will be inhibited.  

The 5101 is three chips to the right of the Game PROM, along the top left side of the CPU board.

Yes. Game prom should be the one with the L

With the machine off.

Put the red lead of your DMM on the upper right pin 22 of the 5101 RAM, and the black lead on ground. You should get 3 to 4.5 volts DC if the battery backup is doing its job.

If voltage is good. Check the capacitor C2 next. Hopefully your meter reads pfd..

Or just replace it. Its 30+ years old now anyway.
A couple of $. And two leads to solider.

Screenshot_2015-03-23-15-45-58.png

If that isn't the issue, maybe you can contact the op in this thread and see if it was resolved.

He had the same issue.

https://pinside.com/pinball/forum/topic/gottlieb-system-1-dip-settings

Another thing to check is if the start button is bypassed/wired to the bottom of cabinet diodes.
That is a free play hack. It can mess up the credits too.