On a separate piece of bristol board, cut a 2.5 x 2.5 inch square. Then cut a smaller square in the bottom left corner. This opening should be just large enough to fit the coin cell batteries (stacked).
Using a pin, poke three holes in the circle. Place one hole in the center and two near the outer edge. Run a piece of uninsulated wire between the two outer holes, so that a straight line of copper wire is formed on the back of the circle. Twist the wires together on the front of the circle (perhaps twist them into a bow or flower shape). This will become a switch to turn the light on and off.
In the image on the left, you can see the circle is in the corner opposite the batteries. Place the circle, and mark where the copper line you made in the last step is. On the paper behind the circle, poke four holes underneath the copper line. Here you will run two more copper lines, perpendicular to the copper line on the back of the circle. The idea here is that when the circle rotates, the copper wire on the circle will cross over the two copper wires underneath, creating a closed connection between the two of them.
I've taped two 3v coin-cell batteries together (in series), and attached the positive side of the battery to the positive leg of the LED, the negative leg of the LED to one side of the circle switch, and the other side of the switch to the negative connection on the battery.