Well simplest method is to use some type of latch. That utilizes two buttons - one for latching the input state and second for reseting it. But of course there are two buttons...
So one of the simplest method is to use NE555 timer, as shown on the schematics:
It works really simple. Reset (4) pin is tied to Vcc, so the IC is not resetting. Discharge (7) is not connected anywhere, so that means the capacitor (C1) will stay charged even if you turn switch off. As soon as you pull the switch, the C1 will charge and stay charged. That will also change the state of NE555s internal comparator, because now you´ll have more than 0V on trigger (2) pin, so it latches the state of capacitor (ON). When you press the button again, capacitor will discharge and now you have 0V on treshold (6) pin, which is not higher than 0V trigger value, so the latch will reset. Of course, you have transistor and resistor on output so you can draw more than 20mA from NE555. Hope i helped :)