I'm very skeptical of this HHO thing as a way of saving gas, however as a 20 year retired AF jet Mechanic I can see the benefits on making an Engine run cleaner and cleaning Carbon build-up, which in itself is a gas user.
Many people may not know it, but up until the turbo-fan jet engine was developed, and normal compression Jet Engine on many of your heavy ACFT used water that boosted performance for take-offs, but one of the side effects of this that has been found is it seems to make parts last longer and burn cleaner, not to say this works on an internal combustion Engine.
One thing to remember, quoted maximum theoretical efficiencies of IC gasoline engines are only good when you put gas and air into them. Put different or extra stuff with different thermodynamic properties and the maximum theoretical efficiency CHANGES. The equation that governs this raises a thermodynamical coefficient of an input to an exponent, so relatively small changes can mean big swings in the numbers. Given that the number in question for Hydrogen is a power of 10 higher than most anything else that goes in, one can see that it wouldn't even have to burn for a relatively small quantity to have a measurable effect. Plus there's a lot of BS about how little mass or BTU of it goes in, well Hydrogen is very light, for it's mass it has a relatively huge volume, under a small amount of vacuum it expands a LOT. In vacuum, the motor isn't really sucking much air and breathes through a straw, literally, there's a small idle air passage in most throttle bodies not much larger than a drinking straw, meaning that in idle and high vaccuum conditions, actual ingestion of air through the idle passage or barely cracked throttle body is of the order of 5 or 6 litres a minute, so stick 2 litres a minute of hydrogen in there and it's a significant quantity, it's a quarter of the volume, then if it expands much more than air in vacuum, it's half the volume in the chamber... again, it doesn't even have to explode to have measurable effects here. When you floor the throttle and the motor is sucking in 1000s of litres a minute, THEN it's not going to have much effect... coincidentally this is what happens on a a typical dyno test when someone "proves" it doesn't work.
I remember The RoadWarrior..To understand who he was, you have to go back to another time..the world was powered by the black fuel & the desert sprouted great cities..Gone now, swept away..two mighty warrior tribes went to war & touched off a blaze which engulfed them all. Without fuel, they were nothing..thundering machines sputtered & stopped..Only those mobile enough to scavenge, brutal enough to pillage would survive. The gangs took over the highways, ready to wage war for a tank of juice