The rest of my electrode materials have arrived. I have not tried all of them, but I had immediate success with a magnesium rod and two sheets of activated carbon. The battery is drawing 500mA when short circuited!
I attached my new battery to a DC/DC boost circuit and an LED array. Both were taken out of a garden solar light. Most garden solar lights use a single 1.2v NiMh cell, which is nearby to the 1.3V that my battery is producing when the LEDs and boost circuit are connected. To the left you can see how the circuit is pulsing the battery input on my oscilloscope:
The battery will work with many different solutions. Even old wine! (although not as well):
An interesting side effect is that the battery will bubble when a load is applied. I might find a way to incorporate this into the final design.
Lighting the water itself might be interesting: