I have a few minutes on my hands right now, so...I'll chime in. I was a mechanic on a submarine for a few years, and we had an oxygen generator that used electrolysis to convert water into hydrogen and oxygen (we threw away the H2, and kept the O2 in a storage tank). We had a lot of excess electricity laying around that we could use for anything we wanted, and the Navy felt that a machine that can make oxygen could come in handy. We normally used a large snorkel to just recirculate surface air once in a while, but...clearly electrolysis works.
Pure water is actually a poor conductor, but tap water has minerals in it that make it conduct very well. For an electrolysis generator, use distilled water and add a electrolyte that you have control over its composition and purity. Salt will produce chlorine as a byproduct (bad), and NaOH/KOH seem to be popular. If you use tap water, you might have to take it apart frequently and clean the crust out of it (hard water has a LOT of minerals). If you are converting tap water to gas, the minerals will be left behind somewhere.
Hydrogen has a wide flammability range, meaning that...it will ignite very easily, and the air-mix (for adding oxygen) does not need to be at a specific percentage. Gasoline vapor needs the fuel / air mix to be close to 14:1 or it won't ignite. Its hard to store hydrogen because it will embrittle steel tanks. If you try an old propane tank, coat the inside with some kind of hydrogen-resistant coating.
Use high-quality stainless steel reaction plates, or...plan on replacing them occasionally. Alternate the positive and negative wires occasionally. Roughen all the surfaces of the reaction plates with 80-grit sandpaper. If you don't separate the electrodes (so the O2 and H2/H2 will be separate), the resulting gas will be a mix, called Brown's gas. Some people call it HHO. It makes a sweet tiny torch (from what I've seen). Use a bubbler/bong as a flashback preventer as close to the torch as possible.
https://www.youtube.com/watch?v=eDXA7Veg6FU
I've heard it works well with 2V DC, some builders claim higher gas production with pulsing. If you are using 12V (as on a car gas system), you might consider constructing six cells in series for 2V each. Configuring plates 4mm apart seems to work well. Father away requires more amps per gas volume, closer restricts gas flow.
Here's my problem with the so called "HHO" car fuel mileage improvers. If you are trying this on a modern car with electronically-controlled fuel injection, along with an electronic ignition...the ECU monitors the oxygen sensor in the exhaust to see if you are getting complete combustion. The government is concerned that...if you are running lean or rich, that you might be producing pollution. Our concern is whether or not you are wasting fuel. They work pretty good, and today, most cars are running with very stable and complete combustion, regardless of RPM. If you are getting a complete burn with no leftover unburned gasoline, there is no extra "wasted" power that the HHO will tap into.
You can run a generator that normally uses gasoline on hydrogen or Browns gas. It will run cleaner, because there is no carbon being fed in. However, it will also have less horsepower. Gasoline is C8H18. It burns the H18, and leaves the C8 behind. Or, it makes CO and CO2 in the exhaust.
IF...you make or buy a Browns Gas (BG) generator using electrolysis. And IF...it produces a lot of gas (liters per minute). It will displace some of the air that is normally drawn into the engine. IF...you adjust the carburetor to run "a little lean" then any fuel mileage improvement will come from running lean, not from displacing air with Browns gas.
BG "can" improve ignition reliability when running lean, especially in an engine with only two valves per cylinder, in a cylinder that has poor swirl characteristics. That being said...under the best of circumstances, you might experience a 5% improvement in fuel economy, at the price of having lower power, and the requirement of filling and maintaining a BG generator.
This is not a theory, many cars have been converted from gasoline to running on propane (C3H8), and the fewer hydrogens the molecule has, the lower the power you'd get. I have heard of V8s that the owner claims it now has the power of a V6, with cleaner emissions. The fuel mileage was hard to quantify, but the fuel costs varied according to the variable price of gasoline vs Propane.
If you have a V8 with a carburetor, and you want to play with this, it can be a fun experiment. If you have a 4-cylinder with EFI and electronic ignition, I absolutely guarantee that if a 25% improvement was possible, you wouldn't be reading about it on youtube (going from 30-MPG to 38-MPG?). These are easy to make, there is no secret sauce. Extraordinary claims require extraordinary proof.