The best electrolyte we found was hydrogen peroxide + table salt. Hydrogen peroxide is available at drug stores and safe to use, but we suspect the solution has to be made up fresh -- it won't keep.
Another battery they talk about uses 2 different electrolytes:
copper metal | copper sulfate solution | salt solution | zinc metal
This uses 2 different paper layers soaked in each electrolyte: copper sulfate and a salt solution. The problems with this is the paper would eventually dry out unless it is in a sealed container. Putting shrink tubing around a penny and zinc battery might help.
But ethanol, ammonia, vinegar, sulfuric acide, and muriatic acid can also be used. But saltwater can be very harsh on certain metals like magnesium.
By looking for a "galvanic potential" chart we can see that aluminum and graphite in a solution of hydrogen peroxide and salt can work well, but the hydrogen peroxide will not last long, especially if exposed to light.
More examples of electrolytes, including bases.