Before even thinking about cleaning a silver coin that you think is valuable, bring it to a coin dealer, who will probably strongly advise against cleaning it. Coin dealers refer to cleaning coins as "whizzing" them, and by destroying the natural surface patina and shading and leaving microscopic scratches, it can render a valuable coin worthless. If you want to clean your old, tarnished coins as gifts or keepsakes, though, you can use several methods.
Soap and Mild Abrasives
Before you start to remove metal tarnishing, you should first remove grime, surface dirt and scuff marks by immersing the coin in a mild soap solution. Fill a bowl with water and add a few drops of dish detergent; then immerse the coin and leave it for several hours, turning it frequently to dislodge material from the metal. After removing the coin from the soapy water, clean off stubborn gunk by making a paste with water and baking soda, which is a mild abrasive cleaner. Immerse the coin in the baking soda paste to cover both sides with it; then run the coin between your fingers.
Acid Bath Methods
The dark tarnish in a silver coin is oxidation, and immersion in an acid can remove it. One method is to leave the coin in a bowl filled with lemon juice or vinegar for up to 24 hours or until the surfaces of the coin are clear. You can expedite the reduction reactions that remove tarnish by immersing some small iron nails along with the coin; if you do, the coin should stay in the bath no longer than 30 minutes. Turn the coin frequently when it's in an acid bath, and if you immerse more than one coin, make sure they don't touch each other.
Stronger Cleaning Methods
You can employ one of several other cleaning methods, but most are more drastic and should only be used on coins with little numismatic value. For example, you can immerse a silver coin for a short period in a strong acid, such as hydrochloric or sulfuric acid, to dissolve oxidized chemicals and mineral deposits that tend to collect on very old coins. Dissolving washing soda in boiling water creates a strong soap solution that removes dirt that has become encrusted on the metal. It's best to avoid abrading the metal altogether, but if you need to clean dirt from soiled edges, use a soft-bristle toothbrush. Never rub a silver coin with metal.
Cleaning by Electrolysis
Electrolysis is the most drastic cleaning method to quickly remove oxidation, and it works so quickly that it looks like magic. Find a flat glass broiling pan much larger than your coin and fill it with a saturated solution of water and equal amounts of salt and baking soda. Flatten out a piece of aluminum foil; drop it into the pan and push it to the bottom. Place your coin on top of the foil, and an electrolytic exchange between the silver in the coin and the foil will clean all oxidation off the coin in a matter of seconds.