Circuit for automatic watch winder
If you want bells and whistles (perhaps literally) you can use an Arduino. You can buy or make a relay interface to switch the motor. If you want it to flash LEDs, have a menacing red LED countdown display, play a song, or coordinate jets of water into the air as your Tourbillon self-winding watches are wound.. it's all pretty straightforward. 

If you just want a silent electronic timer with rough timing but fairly precise on/off ratio, a 555 and CMOS counter (plus a gate or analog switch to determine the on/off cycle) will do it.