You could lathe the stick to be conical, so any practical size ring would fit. Then, take an inch tall section about where the ring would sit, and cut a square chunk out on one side -- depth of the cut would be about 33-50% into the diameter of the cone. Basically like a trigger.
Take that side chunk and drill three inset holes into the inside vertical surface, where it mates against the inside of the cone. Top and bottom two holes would be small springs, like from the inside of a pen. Middle hole is a metal contact. Drill matching holes into the inside of the inset chunk on the cone, to hold the other end of the springs and accept the contact.
Finally, drill a hole through the center of the cone from the base to the top. This is where your LED goes, and the wiring inside. At the point where the hollowed hole intersects (perpendicularly) with the contact hole from the side chunk, cut one of the wires and put two contacts of whatever you can fashion, so when the ring is around the base, the chunk is pressed flush into the cone, and makes contact with the cut wire, thus completing the circuit.
Let me know if you need a diagram.