user33053: When starting the serial monitor in IDE I want to save important data to EEPROM before the Arduino Uno restarts. I do not have physical access to the Arduino so a "press button" solution does not work here.

user4143: You can't. There is no way of detecting if the serial monitor is open or not.
However, you can disable the automatic reset and provide some functionality in your code for saving data to EEPROM when prompted through a command entered on the serial monitor (if you now need to even save the data).

33053: <p>When starting the serial monitor in IDE I want to save important data to EEPROM before the Arduino Uno restarts. I do not have physical access to the Arduino so a "press button" solution does not work here.</p>
 4143: <p>You can't. There is no way of detecting if the serial monitor is open or not.</p>

<p>However, you can <a href="http://playground.arduino.cc/Main/DisablingAutoResetOnSerialConnection" rel="nofollow noreferrer">disable the automatic reset</a> and provide some functionality in your code for saving data to EEPROM when prompted through a command entered on the serial monitor (if you now need to even save the data).</p>
