The Phrase Finder says the expression refers to, “A direct and effortless solution to a problem.” There was an ancient belief that silver had magical powers. There are records of spears and bullets being made from silver to battle evil enemies. Nineteen century fiction has numerous references to silver bullets being used “…against werewolves, witches, the Devil, and all manner of creatures…” The use of the expression for the current meaning didn’t become popular until the Lone Ranger radio show began describing how the hero “…arrived from nowhere, overcame evil and departed, leaving behind only a silver bullet and echoes of ‘who was that masked man’?”  There was a warning published in 1951 about viewing the atomic bomb as a magic weapon. “This is not a silver bullet which can deliver itself or otherwise work military miracles.”