Aside from just the idea that it has to be "something" (maybe lead, steel, whatever), what is it about silver that makes it so special that it can kill a werewolf where other bullets or a stake to the heart can't?
That connection, which is now an un-avoidable cliché, was made up. However, that belief is so entrenched in the mythology's conventions that it has now become an integral part of it.
"Most modern fiction describes werewolves as vulnerable to silver weapons and highly resistant to other injuries. This feature does not appear in stories about werewolves before the 20th century (the claim that the Beast of Gévaudan, an 18th-century wolf or wolf-like creature, was shot by a silver bullet appears to have been introduced by novelists retelling the story from 1935 onwards and not in earlier versions)." http://en.wikipedia.org/wiki/Werewolf
This silver bullet needed for killing makes sense under the Magical Law of Sympathy http://en.wikipedia.org/wiki/, Sympathetic_magic, or associative magic, according to the Law of Contagion
“The law of contagion is a folk belief axiom found in magical thinking which suggests that, once two people or objects have been in contact, a magical link persists between them « http://en.wikipedia.org/wiki/Law_of_contagion:
On the surface it makes sense, however silver (Argentium) is only one of the metals to be associated with the moon, along with any grey or silvery ones like lead, mercury or quicksilver, or Selenium. Its main association with the moon seems to have been because it was one of the only 7 metals found in antiquity and they made them correspond to the planets:
"The ancients believed in a connection between these seven metals and the seven Classical planets. "http://en.wikipedia.org/wiki/Metals_of_antiquity
Besides he proper moon goddess is Selene, not Artemis
“Both Selene and Artemis were also associated with Hecate, and all three were regarded as lunar goddesses, although only Selene was regarded as the personification of the moon itself. Her Roman equivalent is Luna. » http://en.wikipedia.org/wiki/Selene
Note that Luna or Lune is the name of the moon itself is several Latin languages, hence loony, lunatic, lunar, also:
“Greek selene (Lesbian selanna) is from selas "light, brightness (of heavenly bodies)” http://www.etymonline.com/index.php?term=moon&allowed_in_frame=0
Under the old alchemical table of elements, silver came from the moon, gold came from the sun, mercury from mercury, iron from mars etc. Silver was written as a crescent moon rather than Ag.
So it makes sense that a moon-powered creature would be vulnerable to moon-metal.
Silver doesn't really seem to have held much 'mystical' power in history - although, the Greeks knew about its anti-microbial properties (while not knowing about microbes), and this was 'rediscovered' in the middle-ages.
Apparently, silver bullets were used to kill the "Beast of Gévaudan", but this didn't actually enter the stories until the 1930s (purportedly retelling events from the 1760s). This is the first story were silver bullets where used to kill werewolves. It should also be noted that the bullet was blessed, and this may or may not have been more important. The Brothers Grimm did have a silver bullet kill an otherwise bullet-proof witch some time before.
I did find and interesting paper on silver bullets and their practicality (not written by a mystically inclined author), which also covers some 'mystical' properties (without really sourcing where this information comes from though).
A bunch of other 'mystical' sites link silver with the moon - but then a bunch of others don't. Just look up 'magical properties of silver' on wikipedia for the kookiness.
I've always heard that it was because silver is associated with the power of the moon. This page, for example, says: "The pure silvery Moon was associated with the chaste Moon goddesses, Artemis, 'the Huntress with the Silver Bow'." Because the moon holds power over werewolves, so does silver.
Because according to legend, the villager, Jean Chastel, shot and killed the beast of Gévaudan on June 19, 1767; with a blessed silver bullet. The beast terrorized that area of France and killed roughly 113 people and injured 49; 98 of the victims killed were partly eaten. Several large man hunts for the beast went unsuccessful, with many men having shot or wounded the beast with regular ammunition but with no fatality to the beast itself. The reign of terror the beast brought to France lasted four three years. It was only after Jean killed the beast with his legendary silver bullet that the idea of werewolves being weak to silver was brought about. To this day, the beast's true identity or what it was remains a mystery. But during the time, it was considered to be a werewolf and was treated as such. It led to a mass extermination of wild wolves within the region. The legacy of this story and legend lives on today.