The Skeptic Tank

1)
The technology to electrolyze water is extremely simple, any AC or DC power source with both leads submerged in an electrolytic solution (eg water with a little salt dissolved).
(The leads should be of the same metal, otherwise the anode will begin to corrode and the cathode will begin to be electroplated)
As soon as current begins flowing, bubbles appear on the surface of the leads.

1cc (1mL, 1gram) water can be electrolysized to about 1,230cc of hydrogen and about 9,840cc Oxygen. (enough hydrogen to fill a 1-quart bottle and enough oxygen to fill 2 1/2 gallon jugs - assuming the gas is not under pressure)
The ratio of hydrogen to oxygen in water (H₂O) is 2:1, but oxygen atoms have 16 times the mass of hydrogen atoms, so the ratio in volume (and mass) ends up being 1:8

To convert this 1cc of water into its components requires 4.94 watt-hours.

A typical solar panel of about 200 square inches (1 to 1.5 square feet) has a peak wattage of about 5watts.
The little (about 1 square inch) solar cells from radio shack have a peak power of about 0.15 watts.
So the larger panel should be able to produce 1 liter of hydrogen in an hour of direct sunlight, the small panel would take a few days (33 hours of direct sunlight).

2)
The device consists of a tank with three vertical pipes extending upward, like the one shown here:
http://www.fortunecity.com/greenfield/bp/16/electrolysis.htm
One pipe is for the hydrogen release, one of the oxygen, and the third to both replenish the water supply and as a source for atmospheric pressure to keep the water above the electrodes in the other two pipes.

Assuming the oxygen will be vented, two of the pipes will be open at the top, while the hydrogen pipe needs to be capped with a storage device.

Thinking of the hydrogen gas as an energy carrier, the capacitor equivalent would be an airlock valve like the ones used in brewing beer or wine:
http://www.ebrew.com/Products_A/s_type_airlocks.jpg
When pressure builds inside the electrolysis chamber, hydrogen gas will tend to remain in solution until the pressure becomes great enough to push the water in the airlock up, relieving the pressure and releasing about 5cc of hydrogen.

PROBLEMS:
Because the gas has to overcome the pressure of the airlock valve, it will try to escape any other way it can, so the seals need to be completely tight.
'Air-tight' is not even tight enough, as hydrogen molecules are so tiny compared to the other molecules that comprise air (Nitrogen, CO₂, etc.) that H₂ passes through membranes tight enough to contain larger molecules.
[
Side note: This was an issue during the age of airships, when hydrogen was used to fill zeppelins.
The solution was to paint the surface of the skin. The Hindenberg was coated with a paint made of powdered aluminum in a petroleum base, essentially the mixture used to power the solid rocket boosters for the Space Shuttle.
The fire that consumed the Hindenberg was this paint burning, not the hydrogen - only after the fire ate through the skin of the zeppelin did the hydrogen ultimately ignite.
]

This overcoming of pressure is also an issue when trying to use the hydrogen output to inflate balloons.
Standard Latex balloons are elatic, and require significant pressure to inflate.
A Mylar balloon, being non-elastic is a better choice, although it is still not 'hydrogen-tight'

3)
Only problems here.
A) There is no simple way to have the release of hydrogen trigger an electric switch.
One solution would be to place a float inside the airlock that bobs when the gas is relased and physically closes a circuit.
But I don't have the equipment to do this and keep the seals adequately tight.

B) A spark gap requires significant voltage, far more than can be quickly generated with a solar panel.
One solution would be to include a high-voltage (at least 1,000 volts) capacitor and a high-voltage transformer, like one used in a neon sign or television.
An ideal wattage would be around 500, which would take a long time to power with a solar panel - possibly a few months if using the small solar panel.

Another solution would be to have the electrolysis current pass through a high-resistance material that would heat up enough to ignite the hydrogen.
Unfortunately, hydrogen ignites at 500°C (932°F) which would likely melt the resistor and ignite the surrounding materials before the hydrogen began burning.
But that would likely never happen as the current coming out of the solar panel would never be high enough to overcome the resistance required to generate such heat.

A cheat could include a flame or spark source separate from the electrolysis circuit, but this violates the spirit of a self-contained micropower device.

Conclusion:
I'm not going to try to build this.
If I pursue the project in the future, I would start by making the device rely on wall current, which would be adequate to produce high volume of gas as well as provide the spark