I think the servo is the best option, you can screw a piece of plastic or wood stick to the rotary piece of the servo. So that, if the servo rotates let's say only for 30 degrees, it will be enough to push the button as alec_t said.
the switch will be the I/O input in your case, and the led is the whichever inductive component you want to use. use a resistor between the gate and 0V to avoid floating gate. you should also use a diode between the positive and negative terminals of the servo or solenoid. For more info please check the arduino mosfet examples.
For microcontroller, I recommend you to use a low power one, since you do not use more then three or four pins, you can use an ATTINY and program it via arduino.