Actually that looks more like an AMP Mate-N-Lok connector. The connector doesn't look bad, but to properly fix the rats nest you'd need to replace some of the pins. Bob has those seperately, but it's a pretty common connector so you can probably get them locally as well. If you don't mind loosing the quick-disconnect, you could simply cut it out and reconnect everything with butt-splices, you should be able to find them anywhere that sells wiring stuff.
From the looks of things I wouldn't blame the cat as much as whoever did the conversion. It looks like the extra wires were stripped and shoved into the back of the connector till they touched the pins and completed the circuit. Unfortunately from what I've seen such shoddy work is all to common when keeping a game up and running is all the operator cares about.
Fixing it won't be too hard, assuming you know what was pulled from where, but if not you need to find some sort of wiring diagram for the cabinet. Since it's a JAMMA/JAMMA+ game a wiring diagram shouldn't be to hard to track down.