[Ecky]

My thoughts:

By using an external battery you're adding energy to the system from outside, so it will undoubtedly reduce gasoline consumption. Question is whether this is the simplest or most effective method. Consider the extra space and weight a hydrogen electrolyzer takes up, and the efficiency of such a system (~70% cracking the water, ~30% putting it back together in combustion) - you're getting use of around 20% of the energy you put into the battery. And, the amount of energy you can get out of it is limited to how much you can carry in your battery, times the efficiency of the system (20%). There really isn't all that much energy in your typical lead acid car battery.

By comparison, simply offsetting some of the energy generation needed from extremely inefficient alternators (~40% efficient alternator, and ~30% efficient from the gasoline burned, = ~10% efficient use of gasoline) means you're getting almost 50x the "bang for your buck" just using that extra battery you're carrying around to run the car's electronics instead of from the alternator, rather than using it to crack water. Plus you can carry even more battery if you don't have to carry water and an electrolyzer around.

Less efficient than offsetting alternator use, is using that battery to assist in propelling the car through an electric motor. You're still getting ~5x more bang for your buck that way than with electrolysis. However, by hybridizing a car in this way, you also have the other gains of a hybrid system, such as kinetic energy recovery, or the ability to start/stop the engine using electric motor.

Sure, grid-sourced hydrogen can be an improvement, but I think it misses the forest for the trees.

EDIT: Where it COULD make sense, is if you already have excess renewable (free) energy production at home, and you have the inability to sell it back to the grid at a remotely favorable rate or earn credit for it. Assuming you also have the ability to crack water and bottle pressurized hydrogen, using this otherwise excess/waste energy production to create hydrogen to burn in your car is win-win.