One way to check the capacitors in the game is to look at the surface of each capacitor (the top, flat side of the cylinder). If there is any bowing out of the capacitor, they are failing and need to be replaced. Given the pinball machine was sitting for six years unused I wouldn't be surprised if this was an issue in multiple places. That being said, I'm originally an audio engineer, and hums were often related to bad grounding (which is why tightening the screws can help, since in theory the pcb panel is well grounded).
I'm sure there is a proper way to test capacitors with a multimeter, I'm just electrically challenged so someone else can comment on that.