Total cost could be as little as $2.
Step 1: The heart of the mood lamp
The LED contains a red - Blue - Green LED and a minute microprocessor inside a standard 5mm LED package - IT IS SMALL. All you need is to apply 3 volts to the LED to make it work.
Step 2: The cover/display
A 2 AA cell battery holder and a matching battery clip - you can buy or steal one off a dead PP3 9 volt battery.
And of course the rainbow LED (or more than one if you like).