Mega 2560 powered with 7V don't work correctly
The Voltage-Regulator has a dropout-voltage of 1,2V, but in line to the regulator is a diode, which adds another minimum 0,7V (dont know the exact part number, so i guessed the voltage of a Si-diode at ~1A). So you have to supply at least 5 + 1,2 + 0,7 = 6,9V. I presume the circuit doesnt draw a constant current all the time, so from time to time the regulator cant supply the 5V because you sink under the necessary dropout-voltage of the rehulator/diode-combination.

-> Use 8V  or bypass the Diode D1 (with a solder-bridge) and use 7V if its a must and you are sure the input-polarity is always correct.   

Arduino-Mega-2560-schematic: http://arduino.cc/en/uploads/Main/arduino-mega2560_R3-sch.pdf