Had gotten a couple of these gadgets by General Electrics during Winter Olympic Games in Torino 2006 (that's 6 years ago!): no matter how long I had shaken them, I never saw the white LED lit. Time to take a peek inside.
In there I found a well packed coil into which a magnet slides through with the shake movement; a 4x1N4148 fullwave bridge rectifier, a 0.22F 5.5V capacitor and what looks like a white LED. No resistor in series with the LED, so perhaps it didn't survive the first time 5.5V were applied?
I will check all parts with a DMM, but meanwhile I had the usual bizarre idea: what if I turn the coil into a MW resonating circuit and let the capacitor recharge with the environment RF? Otherwise it will become a simple - and smaller - LED flashlight charged via USB. :-)
Update. All parts inside the original flashlight worked. Maybe there was a broken PCB trace, because voltage did not reach the 5mm LED. The supercap and the LED have now been wired to an USB plug making a USB-rechargable LED flashlight. It will be featured on this blog in the next days.