You can test the functionality of the flipper relay by shorting the collector of its driving transistor (center leg or metal tab) to ground. I don't have the schematics in front of me, but Q15 almost always drives the flipper relay on early Ballys. If the relay seems sounds and the connectors are good, replace the driving transistor and its associated blocking diode and 330 ohm resistor.
Also, make sure you have voltage at all 3 lugs of the flipper coils.
Lemme know if you need parts and I'll mail them to you.