I recently purchased some tiny geared motors in the hopes of making a rewind mechanism for a new clock. They are called N20 gear motors and are available with gear ratios ranging from 10:1 to 1000:1. They are incredibly tiny at 12mm by 10mm by 38mm with all metal gears. Prices are around US$10 on Amazon and as low as US$2-3 on eBay.
These are the motors that seem to hold the most promise. The 1000:1 motors are incredibly quiet. I also tried some 120:1 motors that were a bit noisier. Current was measured at around 0.03A at 3V with no load and possibly 0.05A with a larger load.
I am thinking that a small gear can be added to the 3mm end shaft. This gear would attach to the outside of either the minute hand gear or the nest gear closer to the escapement. A simple on/off microswitch could have enough over-travel to trigger the rewind to lift the motor. The weight of the motor falling would power the clock until it needs to be lifted again.