What should I do with a powered switch that seemingly does nothing?
Divide and conquer.
You turned off the main breaker and the switch went dead. So you know it is going through that panel.

Leave the main breaker on and turn off everything else.

If it is still on at this point, then you have something really strange going on - time for a professional. But assuming that does turn power to the switch off:

Turn on the breakers one at a time until the switch comes back on.

This will take a while, so you can also try:

Turn HALF the breakers on and see if the switch comes back on.

If it does, then turn half off and see if it is on or off. Lather, rinse, repeat. If you have 32 breakers, the first test gets you to 16, the second to 8, the third to 4, the fourth to 2, and on the fifth test you have your answer.
As far as what the switch does, is there any chance you have some outside lights that don't work properly? Also look for any receptacles or covered junction boxes (which might previously have had a light on the switch) outside.
Update: Now that you have determined the switch controls a receptacle, there is an easy way to find the breaker if you don't have a helper (i.e., if you have to run between breaker panel and switch/receptacle to test):

Plug a radio into the switched receptacle.
Turn on the radio loud enough to hear it from the breaker panel. Might be best to do this during the daytime :-)
Turn off breakers one at a time until the music stops.

For anyone wondering why &quot;turn off one at a time until the music stops&quot; instead of &quot;turn on one at a time until the music starts&quot;, this gets to the &quot;old vs. new equipment&quot; issue. An older radio has a simple power switch. Turn it on and unplug it and plug it back in and it is guaranteed to be &quot;on&quot;. Many newer radios (like TVs, computers, etc.) have a power switch that does not really turn everything off. A small circuit stays on to power a clock, status lights, etc. For normal usage that's just fine. But the default behavior (sometimes configurable, sometimes not) for these devices is typically &quot;on external power restore go to 'off' state&quot;, which wouldn't be very helpful here.