I have confirmed that removing capacitor C4 will let the board continue to run without resetting when you open the serial monitor window. HOWEVER — This only works if you’re using the FTDI connector on the Mayfly (and a separate FTDI interface device like the Adafruit “FTDI Friend”) to connect it to your computer.
If you want to have this feature when connecting the Mayfly directly to your computer via the Mayfly’s microUSB port and a standard microUSB cable, then you need to remove capacitor C14 instead of C4. Or you could remove both caps and then the Mayfly will never reset on either type of connection.