You could use a PLL (phase locked loop) IC at both ends (transmission and detection). An example is the HEF4046. By selecting the component values, you can set values for your carrier frequency (FM) and the upper and lower boundaries. I'd suggest that on the transmission side, a button is pressed that will apply a high voltage to the input of the PLL IC.
On the reception side you could use a comparator that would activate a relay when a ref voltage is exceeded, ie, when a signal has been sent.
This is a very crude method with no kind of encoding or error checking, but sounds like it should be of help to you. Also this (followed by the letter) will only keep the lamp on while the button is depressed. I'll leave how to have the lamp stay on to you. Think about some kind of feedback arrangement, or the use of a d-type flip-flop. Good luck!