Environmentally safe products for cleaning oxidized copper include lemon juice and salt, ketchup and baking soda. If lemon juice is not available, vinegar makes a good substitute. Baking soda is slightly abrasive and scours the oxidation away, while the other compounds are acidic and dissolve the tarnish.
To clean with lemon juice and salt, cut a lemon in half, and sprinkle it with salt. Rub the salted lemon over the copper surface, and the oxidation dissolves. If a fresh lemon is not available, use bottled lemon juice or vinegar. For heavy oxidation, it is sometimes beneficial to add cornstarch as a thickener to the solution, forming a paste. Spread the paste on the surface, and leave it there a few minutes to dissolve oxides.
Ketchup uses the acidity of the tomato to clean the oxidized copper. Spread a small amount on the copper, and allow it to work a few minutes before rinsing away the ketchup and tarnish.
Fingerprints on the clean copper cause tarnish to form again quickly. Storing the shined object in a zipper-seal bag or a tarnish-free cloth can help to prevent it from forming again. A coat of lacquer is also useful in keeping copper free of tarnish.