I have though of doing this for my house in the past.
Before I start building the tower I wanted to have a feel for how much energy one can store using this method. I live next to the sea so water is no issue.
Say to power 1 low energy 20watt light bulb for a few hours how large would the tower have to be and how high from the ground.
I'd really appreciate a link to a resource explaining the maths behind this.
I am going to assume 100% efficiency for simplicity. [1]
mgh Joule = 20 Watt * 4 hour * 3600 s/hour
(Joule = kg * m^2 / s^2 = W * s)
Lets say you pump water to a height of 10 meters.
m = 20 * 4 * 3600 / (9.8 * 10)
m = 2938.77 ~= 3000 kg
So you need about 3000 liters (1L = 1kg for pure water) of water at a 10m high tank to power a 20W bulb for 4 hours. This about 800 US gallons. Considering other inefficiencies, you'll need more.
[1] http://www.jcmiras.net/jcm/item/93/ (first google result for efficiency of a generator) says typical efficiency of a water generator is 0.95.
That's about the capacity of a small 12V lead-acid battery that you can carry in one hand. Weighs about 4 lbs. I use one to provide LED lighting in my chicken coop because it's too far from the house to run AC power out there.
No contest: in the absence of extenuating factors, a cheap battery is better than the "innovative" (?) water storage method for such a small amount of energy.
Maybe it would be viable if you lived next to a convenient cliff or other ready structure to simply place a big tank on top of with a pipe hanging down but if you actually have to build the structure yourself it sounds like a non starter. And that's without even considering the other engineering hassles you'd have, like corrosion, filtering seawater, maintenance, etc.
You probably wouldn't even need to break out of the mass * g * height = energy equation.