well, you made a battery. sorta. moreso probably a capacitor.
a very weak one.
two dissimilar metals, in an electrolyte, createing a voltage potential. rainwater has some interesting properties, no doubt, but basically the tiny amount of acid in the water set up a small amount of voltage. the screens you used were probably very similar, which is why they put out so little electricity, but still had a voltage potential.
now on the other hand, the water u were using has more potential as an insulator than an electrode, so hitting the two electrodes with 10V and "charging" the capacitor, then hooking it up to a load as tiny as a solar calculator would explain why it worked better than ever, and then faded out over five minutes.
batteries and capacitors are similar with some very different characteristics-
but either way, id say if you added a bit of lye or even citric acid (try a cleaning agent with anhydrous citric acid) will increase the electrical conductivity of the water, and then put as small of a voltage as possible and as high of amps as possible (lower the resistance of the water and use as large electrodes as possible) and you'll be making enough HHO to (argueably) boost a car.