I would like to point out that the copper electrode used here should not slowly dissolve, for the same reason that neither sodium nor chlorine (from the salt) are released at either electrode. IIRC, it's to do with the electronegativity of each ion involved.
Would someone who's got a better grasp of inorganic chemistry please help explain this? It's been too long since I've tried these electrochemical equations, even for something as simple as this ...