Push button to toggle servo direction - monostable 555 circuit?
A switch is good alternative, whichever would be easier.
I've been looking at circuits to do with 555 timers and servos and have found these two which seem useful:
Is that second circuit what I need? How would I change the direction of the servo each time it runs?
I looked into flip-flops such as the 4013 which came up as a suggestion instead of a 555 but quickly got out of my depth. I'm happy tinkering around with electronics but unfortunately don't know too much of the theory...