first we need to figure out the switch. By taking your multi meter and setting it to ohms we can see which 2 of the contacts are switched. Put one side of the meter to one terminal and one side of teh meter to another terminal. Push the button a couple of times and see if the impedence goes from open(often displayed as 1-----------.--) to showing a short or near 0 ohms. If you get nothing from the first 2 terminals then swap one side of the meter for the other terminal and repeat.
The 3rd terminal should be to light up the switch, and will either go to a ground or constant. unfortunatly we do not know if the switch is made to switch a + or a ground but we will find out when we get it to light up. Try grounding the 3rd terminal and then connect power to the other side, if nothing happens then reverse it, you can probably test all of this with a 9v battery the light will just not be as bright. When you get it to light up, then test the terminal that we found to be common with the terminal above to see if it is sending a + or - signal to start. I would imagine it will send a ground, which means you will need to wire a relay but we will cover that bridge when we get there.