I suspect that the big problem with this concept (vs. the hydroelectric example) is going to be the amounts of energy involved. Let's suppose the shaft is 500m deep and a 100kg battery -- potential energy is simply F * z, so F = 980N and z is 500, which gives you 490kJ. A wH is 3.6 kJ (you'll need ~10-20 of these to run an iPad for an hour). You're going to need a really big weight and a really big shaft.

This shouldn't be surprising -- with a fairly small motor and good gearing you could raise a remarkably large weight a remarkably long distance with relatively little energy.

Also note that with a hydroelectric dam, the storage mechanism also happens to be the power generation mechanism (dam + turbine) so you aren't incurring huge additional construction and maintenance costs relative to just building the power source, so even if you replaced the weights with a giant tank of water, the entire storage mechanism is extra capital investment and maintenance on top of the solar panels.

If the fundamental problem you're trying to solve is baseline power, it's probably more efficient to bring power in from somewhere the sun is shining (8000 miles away, say):

"As of 1980, the longest cost-effective distance for Direct Current transmission was determined to be 7,000 km (4,300 mi). For Alternating Current it was 4,000 km (2,500 mi), though all transmission lines in use today are substantially shorter than this." (Wikipedia)