The silver bullets serve several purposes for the Lone Ranger. First, they are his calling card. Even in disguise the Ranger can be identified by friends when he presents a silver bullet. Honest lawmen know that help is near when they find one. Likewise, criminals know they are in trouble. Secondly, the silver bullets are a symbol. They serve as a reminder of the preciousness of life, and the high cost of pulling the trigger.
In 1981's "The Legend of the Lone Ranger" the silver bullets are explained differently. Tonto presents the Ranger with silver bullets informing him that silver "flies straighter" and is "more accurate" because it is "pure." There is never a mention of a silver mine in that film.