Although I think the idea isn't workable (energy density is too low, cost of boring the hole is tremendous), most of the other commenters here seem to think they'd just have one weight, whether it is 1000 KG or 50,000 KG.
Any sane plan would be to have more than one weight. When the first weight hits the bottom, it would release from the cable and another weight up top would grab the cable and start dropping. To store energy, the top weight would get winched up, and when it hit the top it would lock into place somehow and the next weight at the bottom of the shaft would engage the lifting cable, etc. The cable would have to follow a circular track, rather than having 1KM of cable for each weight.
What advantage does this provide over combining all your weights together into one big one? If you want to limit the tension on the cable, just use more cables.
You shouldn't be so confident that you know the only "sane" way to design such a thing. It's not a field anyone has experience in. These are only guesses.
Well, it does mean your cables and dynamo can be smaller. But then you've got a new problem of coupling the cables to the load at the bottom and somehow storing loads at the top that the cables must reach past/through.
You have a shaft 1km or 1.5km deep. You can afford to stack a lot of weights vertically. Imagine each has a hole running through the middle, where the cable runs, and a mechanism whereby the weight can engage/disengage from the cable. The cable forms a circular loop that passes through all the weights from the top of the shaft to the bottom, then circles around past a large pulley and back to the top, completing the loop.
At the top of the shaft there would be hefty prongs that retract when the weight needs to start dropping, and when the weight returns during a recharge cycle, the prongs reinsert themselves in to the shaft when the weight is lifted back into position.
Most likely each weight would have a "C" cross section with a nearly closed mouth -- just a slot from the edge of the weight to the larger central opening so that the weight can be removed/replaced from the cable if needed.
My understanding is that peak power availability is at least as important as total energy storage. (The main use of grid storage is to handle spikes in usage, or similarly to fill in for dips in generation - like from wind turbines. Either way, peak power available is important.) One big weight would give you significantly more peak power than several smaller ones. Yes it would require more or stronger cables, motors, and other infrastructure, but the cost of all those is small compared to the shaft, so I imagine a single weight would maximize peak power / cost.
That's a good point. Say you fill half the hole with containers of water. Let's assume it's 1km deep. Say we have a circular hole that's 2m across, that should get you W = m * g * d = (3.14 * 1m^2 * 500m * 1000kg/m^3) * 9.8m/s^2 * 500m = 7.7 * 10^9 J or 7,700 MJ? That's a lot higher than the numbers being thrown around here.
Edit: just noticed the comment about 500,000 kg -> 1.2MWh. Read that as MJ at first... d'oh.
Any sane plan would be to have more than one weight. When the first weight hits the bottom, it would release from the cable and another weight up top would grab the cable and start dropping. To store energy, the top weight would get winched up, and when it hit the top it would lock into place somehow and the next weight at the bottom of the shaft would engage the lifting cable, etc. The cable would have to follow a circular track, rather than having 1KM of cable for each weight.