The AVR is powered from a 7805 regulator which takes input from the 555 timer (that has a limited output current capability). Because of that the regulator is not able to supply much current and when the servo tries to rotate and pull current the voltage drops and the mcu resets.
To solve the problem you should feed the servo from a separate regulator that takes input from the 9v supply (assuming the servo can't be powered directly with 9v).
Adding to what Nick Alexeev says, why do you need the 555?
You can use the AVR only, put it in sleep mode and set it to wake up with an external interrupt (INT0) from a button press, work for 10 sec and then sleep again.