Actuating a relay (either 5V or 12V) with 3V
This can boost 3.7 to 12V, about 1 to 2A. Enough to drive sizable relay.

Not sure if relay is needed. Can simpler mechanical switch do?

http://www.coolcomponents.co.uk/adjustable-4-12v-step-up-voltage-regulator-u3v50alv.html#.