Looking to try the effects on:
Vibrations
Acoustic (e.g. running experiment on top a water proof subwoofer)
Super sonic (e.g. running experiment inside a ultra sonic cleaner)

Light
Lasers
Various wavelengths light

Other electromagnetic
Sinewave/ other signal generators using various frequencies to power the electrolysis it's self.
Magnetic fields

Catalyst
Electrolyte concentrations
Various chemical catalyst
stirring the water.

"Life will find a way" (a bit extra mad scientist)
Using diatoms in a salt water electrolysis experiment. Developing some sort reward system that feeds the diatoms as hydrogen is produced. More hydrogen, more resources. With the hopes that over generations of experimentations, these sylica based life forms will evolve some method to improving the process.

Simplest way: every X units of hydrogen produced = food drop.