carbon electrodes have potential difference even with no electrical input?
Some possibilities:


Porous carbon electrodes do act as a capacitor, the basis for supercapacitors (aka ultracapacitors). 
You are electrolytically depositing copper on the negative electrode when you run current through the solution, even though it might not be enough to see. This creates an electrochemical cell.
Simply disconnecting the power supply leads will leave a high-impedance voltmeter floating, so it might be detecting random static charges. You might need to bypass the voltmeter with a high resistance (10 or 20 megohms) to reduce static sensitivity.