We can lay this at the feet of one man, the legendary screen-writer Curt Siodmak. Tapped by Universal Pictures to quickly come up with a script for a B-movie, The Wolfman, he used all his recollections from his native Germany, reviewed old Universal wolf movies, and likely tapped into the ground-breaking novel ”The Werewolf of Paris” (1933) by Guy Endore.

Even so, he came up short and had to create many new iconic symbols. After all, though this was a B-movie, it was going to be a breakout vehicle for Lon Chaney, Jr, and had a number of stars in it. Siodmak was constrained by strictures handed down by the Hayes commission, so no overt gore, sultriness was OK, but no nudity or sex. Evil had to be punished in the end, and he had to make Chaney liable while still eating women alive – not an easy task.
So he invented the shtick about the night activating the feral side of Talbot (with maybe a sly tip to Chaney’s alcoholic binges or the Nazi’s which Siodmak fled), pentagrams in the palms of victims’ hands, agonized spells of guilt, memory losses, a clever poem, and most of all the wolf’s vulnerability to silver.
It might be of interest that the poem which Siodmak created for Wolfman (1941) read:

Aconitum carmichaelii, wolfbane that blooms in Autumn
“Even a man who is pure in heart,
And says his prayers by night,
May become a wolf when the wolfbane blooms,
And the autumn moon is bright.”

The last line was changed to “And the moon is full and bright.” in Frankenstein Meets the Wolfman (1943), thus changing the original autumnal multi-night transformation of Talbot into a more frequent, but ONLY full moon transformation (which usually lasted two or three nights, an astronomical impossibility).

Alchemy symbol for silver
So Why Silver?
Siodmak’s interviews were never overly specific on silver, but apart from the “silver-screen” being a bi-product of silver colloidal chemistry, the roots of silver mythology lay in alchemy.
Alchemy is usually recognized as having two simultaneous purposes – the transmogrification of base materials into gold and finding the “elixir of life”, essentially a “fountain of youth” formula. It was really a very early method of understanding the nature of chemistry based on writings handed down by Greek, Latin, and Arab researchers. The ancients noted that there were seven known heavenly bodies, seven essential and useful metals, and that the calendar could usefully be divided into about 52 “weeks” of seven days each leaving 13 weeks to coordinate to four seasons in one solar year, and a roughly equal to twelve, 29 day, lunar “months”.
Therefore, alchemists and scholars lined up the seven metals, seven days, and picked seven notable deities to be their patrons. In English, the first day was the greatest – Sun-day, ruled by the Sun, which glowed the color of gold, the rare, imperishable, and always bright yellow metal. It’s opposite, the next day (Mon-day) was the Moon, Luna, the goddess of the night and all it’s dark terrors – owls, wolves, bats, and so forth. Silver, slightly more common than gold, mimicked gold’s malleability, but it had the demon in it. It quickly tarnished, especially when exposed to noxious brimstone (sulfur).
Silver was separated from lead (another powerful and important metal ruled by Saturn or “Satur(n)-day”). The word ‘silver’ comes from the old English term ‘seolfor’ – and roots as far back into Indo-European as can be traced, essentially meaning shiny and white. The symbol, Ag, is abbreviated from the Latin ‘argentum’. The metal has very useful alchemical properties that must have captivated poets imagination – as in “by the light of the silvery moon”, or “silvery stream”. Nearly inert to the human body, though it often turns skin green, it was used for drinking vessels after lead was discovered to be poisonous. It conducts electricity like copper does, though it was not until after the 18th century that this became important. It can be beaten, thus it is malleable. Before casting, silver was purified and then beaten into shapes.
However, it bedeviled metallurgists, as silver quickly turned black especially in the presence of brimstone (sulfur dioxide fumes) or rotten egg smells (hydrogen sulfide) both known elements of Hell. When handled a lot it turned hands black as the devil himself. All of these things together, identified with darkness, the night, creatures of the night, and the Moon, mistress of all things of the night.
It was natural, then, that Siodmak turned to silver, the Moon-element, to be the nemesis of the werewolf, the creature of the night. Perhaps Siodmak recalled Balderston’s phrase from the Dracula play as the vampire listened to wolves who created “music of the night”. So Siodmak placed a silver wolf’s head walking stick into Chaney’s character’s hands early in the movie, he kills Bela with it, and later a silver bullet kills Talbot. All under the silvery moonlight, though the Moon itself is never seen.

(Curt Siodmak was a reporter in Germany, and when he interviewed Fritz Lang about his movie career, Lang put him into Metropolis jump starting his movie career. )
