Need a very low power, low force motor/actuator
Here is the drive mechanism from my beloved Butterfly Livingroom Flyer...


It is a pager motor connected to a reduction gear. By picking the size of the gears (or possibly adding additional gear stages), you can have pretty much any viable torque/power combination you want, down to single digits of mA. 

If you need even a smaller actuator, Plantranco also sells the MicroAct Magnetic Actuator - 0.4g...



You could also make something similar to this yourself with some very high gauge enameled wire and a small super-magnet. This would give you lots of control of the power/torque/range-of-motion trade offs. 

(Note that I have no connection to Plantranco other than I have bought more of their flyers than I'd care to admit!)

I also love Jim's suggestion of hacking a clock motor if that could work for your application. These things use impossibly small amounts of current. 

Here is a product that is basically a clock motor, but good for hacking because it comes with lots of possible gear combinations, and also provides continuous rotation (the yellow metal disk acts a a flywheel), whereas clock drives usually have unneeded timing circuits and often move in a a once-per-second jerk ...



They cost about $30 on Amazon, much cheaper if you need more than about a dozen.