If you want the lowest power device, then use a naked ATTiny85, and run it from 3 volts or so at 1MHz. It will only need microamps to run 'every now and then', while sleeping most of the time..
As Robin2 suggested, you need to decide what motor you will use. The servos from your drones will work, but servos are power-hungry. Perhaps you could run the microcontroller from one circuit, where the battery would last for a year or more, but power the motor separately (which is recommended anyway.) If you have a multimeter, hook it up as an ammeter, and measure the current needed to start, and to sustain the motor. Then calculate how long they will last on a single battery charge. Perhaps consider adding a battery charging circuit, either plug-in, or solar if you are off the grid.
ps, Robin, "super-abstemious"... You sir, are a vocabulary god!