If you have experience with Arduino, you have most of the project in good stead. Servo motors and stepper motors are commonly used with Arduino to accomplish similar tasks. Your answer lies as much in your budget for this project as it does for the selection of motors.
A 360° capable servo will rotate when driven by the proper code, operated by an appropriate servo driver board. Non-360° servos are limited to about 180° movement, far too little for your rotation needs. You can find code libraries to cover this aspect, as well as appropriate hardware from the multitude of Arduino sources, Sparkfun, Lady Ada, Amazon, etc.
The other option would be a stepper motor, stepper motor driver boards and code libraries. As you've noted that a bit of overdriving isn't a problem, you can get away with some slop in the positioning. A stepper motor is commanded to move a specific number of steps. If during this command execution, movement is impeded, the motor keeps counting and may stop short. Some stepper motors also have encoders to report location/progress, but that increases the price, of course. Servo motors by design "report" location and attempt to move until success or servo motor burn out.
Stepper motors would have sufficient power, even those among the low end force models, while servos would require purchase of the higher end components in order to get working devices.
Mechanically, I suspect you'd have an easy time of connecting either type of motor. A housing/bracket for the motor, a coupling to the rod, even a simple piece of vinyl or rubber tubing, and you'd be ready to go.
I expect the difference between the price of parts required to create nine sets of drives and the price of nine factory-motorized blind sets would be substantial, in your favor.
As a DIY project, it also would provide you with a sense of accomplishment. What value can one place on that?