How to start LED wirelessly in a very compact and cheap module?
I would use something like a Wixel.  It combines a low-power MCU with a radio.  It may be possible to combine a low-cost MCU with a low-cost radio (for instance, an nRF24L01) to achieve a slightly lower cost per unit, but the development time would be higher.

There are also Bluetooth LE modules like the BlueGiga BLE112 that may work well.  The last time I looked at it, the BLE112 seemed a little harder to program than the Wixel (more complicated development tools), but the cost per unit is lower.