Which GPS? Is it rechargeable via a USB computer connection in addition to disposable batteries?
It's relativity easy to recharge Ni-MH batteries as you pedal along by using a dynamo. All you do is use a bridge rectifier circuit to convert AC made by the dynamo to DC for the batteries. Ni-MH batteries will regulate the voltage of the dynamo automatically. As voltage approaches the Ni-MH batteries rated recharging voltage internal impedance created by the batteries will prevent the dynamo from being able to produce more voltage than the batteries can stand. This allows from between 2 to 5 1.2-volt rechargeable batteries to be connected to the bridge rectifier and recharged as you pedal along. You can have one set of batteries in the GPS unit while the second set is being recharged. You will need to calculate how much time it will take to recharge your batteries. This information is located on the batteries. Nearly all dynamos provide 500mA which just happens to be the suggested rapid recharge rate for Ni-MH batteries. After you figure out how long to recharge the batteries simply reset the trip time on your cycle computer which will start and stop as you do and when it reads the time required to recharge your batteries just disengage the dynamo. Here is a photo of a bridge rectifier using 1N5818 diodes soldered to a terminal strip.

The yellow wire attaches to the dynamo and the other dynamo connection is grounded to the frame of the bicycle via the mounting tab. DC output is via the red positive and black negative wires. There is no need to water proof as it's a solid state device immune to fresh water.
Here is the circuit I use to recharge my GPS unit or a cell phone via a USB connector.


I am adding a better pic of the bridge rectifier to make it easier to copy.