I think the largest issue you’ll run into is figuring out how to wire it. It definitely sounds like a fun project and possibly an opportunity to try out some of the inexpensive super caps that you can get on eBay for like 3 bucks a piece.
The only caveat I can think of is that you might need to find out how much voltage the generator puts out, as you definitely don’t want to pump more voltage into them than they are rated for.
To do that I would open the device and hook up a multimeter to the Dynamo and then spin it as hard as I can and figure out what it’s peak voltage is and then purchase and or design your super capacitor bank to handle its peak maximum.
You should also check and see if the device has some sort of rectifier, as Dynamo output is typically AC and LED lights are typically DC, so there is likely some sort of rectification. If you do have a rectifier I would measure it after the rectifier.
It may already also have a capacitor in it, in which case you could simply desolder that capacitor and then set up some breakout cables and run those outside of the case into your bank.
If I’m understanding it correctly this generator has like wires out that then connect to the lights or something right?
If that’s the case you’ll need to see what kind of voltage it’s putting out to those wires and then you would want to splice those wires into diodes to prevent the current from leaking backwards out of the capacitors. Then attach as many capacitors as you need for the voltage that it is providing, remembering that it is okay to have higher voltage ratings but not lower voltage ratings.
Chances are this will cause the light to be less bright as the capacitors are charging, so for the first few moments when you are riding it probably will be less bright but after the capacitors reach a steady charge you should see the light take a much longer time to dim down.