If several batteries are connected in series while performing electrolysis, how would the rate of electrolysis vary? Why?

When you connect several batteries in series, you are increasing the amount of voltage passing through the circuit.  The water you are using is also part of the circuit the electricity passes through.  Since there is an increased amount of voltage, it would be reasonable to expect to see an increase in the amount of hydrogen (H2) and oxygen (O2) gases produced.  For example, a normal electrolysis setup would involve one 12 volt lantern battery.  The amount of voltage passing in the circuit would be 12 volts, which would give a normal (n) amount of hydrogen and oxygen.  If you wired two of these 12 volt lantern batteries together in a series wiring scheme, you would effectively double the amount of voltage, 12V  +  12V  =  24V.  You could expect to see twice as much hydrogen and oxygen gas produced, because you doubled the amount of voltage in the water.