I want to add a driver to a vertex' z position. Is this possible? If it is, how?
When I try to add a driver, all I get is the option to edit source and edit translation.
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You could also add Shape Keys and animate the influence, either by adding keyframes directly or by a driver.
To answer the question specifically: Yes. It is possible.
Go to Outliner, switch Display Mode to Datablocks, then you'll see a list of all datablocks for the .blend file.
Expand the list: Blendfile Data > Objects > your_obj_name > Data > mashdata_name > Vertices > Mesh Vertex > Location, then you can right click on X Y Z, and choose Add Drivers or Add Single Driver from the menu.
However, when there are too many vertices, it will be very hard to target the one you want. I would rather use hooks.