It can be done as a DIY. You'll need a small momentary normally closed switch. When the lid is closed, it'll push the switch in and turn off the LED. As for the LED, you should get a right mount LED to make it face towards the front of the car so it doesn't blind you at night when opening the cubby holder.
As for wiring the LED, your best bet is to find the DIM wire in the harness so the LED only lights up when you turn on the head lights. As for the resistor (for std 5mm LED 3.2V @ 20mA), use at least a 480 ohm resistor.
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