That's a tough one. From it's position it looks like something they don't anticipate common use so it's probably not for overhead lights. What's in the black area right above the switch in the picture?
It will most likely take some good old fashioned sleuthing skills. It could be an exterior light. wait until dusk or dark and walk around inside and outside and make sure no lights are on. Then switch the switch on and walk around again.
Turn off the water heater and let the tank cool (maybe turn it off the night before you go to bed) In the morning flip the mystery switch on and check the water temp after 30 minutes.
Is there an outside entertainment system or plug that it might control?
Could it be for the water pump like for if you forgot to turn on the pump before heading to the bathroom at night?
Are the fuses in the breaker box labeled? If so try turning on the switch so it lights up. Then pull a fuse and see if the light goes out. If the light stays on replace that fuse and pull the next one.