The differences in using other metals or materials would be compounds they from as sulfates (solid or soluble, dissolved), how much energy it would take to split their bonds, and how quickly they would react. Let's see..
For copper, starting with Cu2O. It would take three times the amount of Cu2O to soak up the same amount of O2 as an iron cell (Fe2O3). Copper sulfate is soluble, and when in contact with water, exists as copper sulfate pentahydrate, absorbing 5 water molecules per CuSO4 molecule. Each Cu2O molecule will be able to react with 2 sulfate molecules, which is the same as iron. This makes 2H2 molecules and one O2 molecule for each reaction. Nice. Problem is, that I've seen, iron oxidizes very slowly, even exposed to electricity as we plan to do. I don't think a recharge cycle of a few hours for 450g (a pound) of iron is very practical. I think copper will do better, but marginally so. I don't like to redesign things like this, but it may be in order.
For CuO, another stable form of copper, We'd have half the hydrogen production of iron for the same molar weight (1 mole), so we'd need to double the amount of CuO we use, which would also put our H2 vs. O2 production at 2:1, standard electrolysis levels.
For zinc, which is thankfully more reactive than the above, and slightly more reactive than iron, oxidizing faster than the ones above, would have the same results as CuO, but possibly a faster recharge cycle.
For lead, with its more stable 2+ oxidation state, we would again see the same results, but as it's less reactive than any other metal here, except copper, a slow recharge cycle could be a problem.
Our iron will rust very slowly as well, unless we can very rapidly expose it to a lot of pure O2 (doable, seperated and bubbled through, but it makes the rest of the process less efficient, since part of the products is put bck through the cell. I like the idea, but it's impractical.
