It's not going to be a complete answer (at least not yet), but I'll try to explain the idea. I still have to work more strictly on details.
First let me be very specific about one thing. I'm handwaving here the difficulty of building so tall mechanism and probably few other things. This is just an idea how things could be organised.
First of all, as others already mentioned, we need a setup with a counterweight and to make the trip possible the counterweight needs to actually adapt it's weight to the weight of a cargo (be it human, weapons, building materials - doesn't really matter).
As I imagine it, counterweight is liquid. Since the temperature is below water freezing point, the liquid used has to be either some kind of solution or other liquid (e.g. alcohol) in order not to freeze.
At both top and bottom station there would be a mechanism that enables balancing the counterweight. As soon as the counterweight sets itself in the correct position (close to the stop), the fluid starts to flow in/out until it gets exactly even with the weight of the cargo, making the elevator stop. 436
Once stopped the platform and counterweight are locked in their positions (i.e. can move slightly but only within specific very limited range). Whenever the weight of the two becomes unbalanced the shifted position of counterweight causes the flow of liquid in the right direction causing it to automatically balance the counterweight against the platform with cargo. When a lever is activated, it unlocks the movement of elevator and at the same time closes the counterweight liquid tank. Due to a small difference in mass (potentially with some speed limiter) the elevator starts to move in the right direction.
Depending on the design there are two options:
1. The lever activation adds or removes a bit of liquid to the counterweight making the movement possible, or
2. The counterweight filling systems are scaled so that it becomes slightly heavier at the top and slightly lighter at the bottom.
And that's pretty much it.
The system might require from time to time balancing the amount of liquid in top and bottom tanks but it might be scaled so that this maintenance isn't needed too often. 436