It may have to do with the Lunar Lattice: at least in some Earth mythology, silver is the metal of the moon, which is the source of its power against the unnatural (who apparently are also moon-powered). So it's a case of fighting fire with fire.
Besides the affection of silver to Lunar "influences", it was also widely used to fabricate mirrors. Throughout historic literature "mirrors and ghosts" are referenced abundantly, and in many myths (even current practices), if a ghost is to catch it's own reflection then it will be 'turned' from haunting.
A bit of a long shot, but what the hell.