Instead of morse code you could change the brightness/color of the led depending of the time. In the evening the led could start to turn darker and in early morning it could start turning brighter again.
Usually you don't need to know exact time when you wake upp in the middle of the night, therefore an approximate time would suffice.

You could even add some gimmicks, when full moon the led could blink yellow from time to time or maybe it could blink according to the weather?