Yes there will be a separate power feed for the light and power to the lighter itself. As there are 3 wires, I would guess at 2 12v+ feeds and a common earth.
Set the multi meter to DC volts, mine for instance has a setting for 2v, 12v 200v, I would set it to 12v. Connect the black wire of the meter to a known earth point (any metal part of the car body - not painted, for instance the door striker) or the wire on the plug thats the earth - if you are lucky it will be brown or black!
Connect the red wire from the meter to the connections on the plug, you should get a 12v reading from 2 of the wires, if only one reading then you have a bad connection.
Hope this helps.