If you want it to light all the time, as soon as the game is powered up, check the bulb to see if it is 5V or 12V and hook one end up to that lug on the power supply, and the other end to ground.
If you want it to light up when the game sends a signal to light it, you'll need to find the correct wire in the harness, and the other end goes to ground.
I have about 20 of these Big BUck Hunters, and none of them have lighted buttons though, so I'm not sure if yours is different than mine, or maybe yours was originally a different game that was turned into a BBH, and the bulb and socket are left over from the old machine wiring.