Sounds like fun. I can't remember, but maybe it was old telephone backup that stored power in a big gyro and used it when power failed.
Is this just a demo or is it important that it be highly efficient, etc.?
Some things to consider:
The more power (bigger battery) the faster or heavier the gryo, so it might be good to try to match things up.
If you need to squeeze all the power out of the gryo the voltage could be boosted once it fell below 6 volts - but more complicated.
If efficiency is important you would want to charge the battery as quickly as possible to avoid wasted power in friction. -- also a bit more complicated.
The same is true. If efficiency is not important an analog charger would be simpler but a switch mode more efficient.
Any idea of the energy stored in the gryo?
Can you use purchased assemblies or must it all be home grown.