We are told that Arfaja al-Bariqi's nose was cut off in battle, and he had a replacement of silver made; but it became 'offensive', and Muhammad told him to make a replacement out of gold, which would not become spoiled.1
Ambroise Paré (1510-1590) recounted a tale of Cadet de St. Juan, who, finding his silver prosthetic nose tiresome, went to southern Italy and received a state-of-the-art surgery to have a skin graft to repair his nose. The date of the surgery is not recorded, but the time period is consistent with the golden age of Italian reconstructive surgery.2
In 1566, Danish astronomer Tycho Brahe lost his nose in a duel; he replaced it with a false nose made of gold and silver.3
Many of the web-pages writing about replacement noses, and especially silver noses, will include an image from the London Science Museum, identified simply as a "plated metal artificial nose, 17th-18th century", from "Europe". I am not convinced that it is silver plated, but the internet wants to believe.
The Hunterian Museum of the Royal College of Surgeons has a prosthetic silver nose on display, dated to the mid-1800s; it is attached to a pair of eyeglasses and has straps going over and behind the head, ensuring a good fit. It was likely to replace a nose lost to syphilis.
In 1876, we are told "during the war between Russia and Turkey, many of the Sultan’s soldiers had their noses amputated for revenge by the Bulgarian invaders. Quite soon, thereafter, the enemy having been driven back and overcome, the Turkish Sultan made a gift to each of those mutilated, rewarding them with a nasal prosthesis in pure silver which they displayed with pride in the streets of Istanbul, like a decoration of great value"4
Since at least the first century CE, and possibly earlier, Indian surgeons have been able to repair lost noses with skin grafts; by the 1500s, the Italians were able to repeat this feat, although others were not successful. Those lacking this technology have made false noses from whatever was at hand: leather, wax, clay, wood, paper mâché, and metal have been used, among other substances. But over and over again, the noses that get the press are silver noses. Which is odd, because silver is heavy, expensive, corrodes, and offends the eye of Muhammad.
Attentive readers will have noticed that, of the six examples cited, only four actually are actually "probably silver"; one easy answer to the question is that perhaps many noses reported as silver... weren't. While there are certainly more stories of silver noses than those listed above, those above were the ones that were easiest to research; other threads proved even more tenuous. It is quite possible that many noses mentioned in passing as silver were in fact some other shiny metal.
However, it is also possible that silver was a good choice; while it corrodes, it doesn't 'bleed' the way copper does; while it is heavy, it's nothing like as heavy as gold. And if you must have a false nose, surely, that's a good thing to splurge on? After all, more people will be looking at your nose than your rings or cuff links. It is also antimicrobial and fairly hypoallergenic (although perhaps less so historically than today). Silversmiths were also common enough, and used to fiddly work; a tinsmith might not produce such an artistic result.5
Perhaps, though, the single most important reason to have a silver nose was to show that you were wealthy. This is always a good thing, of course, but historically, people often lost their noses as a form of punishment, either for breaking a law or for losing a battle. A wax nose might not do much to dispel the possibility that you are a common thief (or footsoldier), but a silver nose makes it quite clear that you are not currently listed among the losers. This also explains the willingness for rich people to undergo early elective reconstructive surgery, which was risky and extremely painful, and we must assume, unlikely to result in an attractive outcome.
There is, though, probably a much stronger reason that we hear about silver noses. A wax nose or a leather nose simply isn't impressive. Even an openly deformed nose doesn't particularly stand out among the oddities of history. A silver nose, though, makes a good story, and one that will be passed down through the ages.
Footnotes:
1. These tales only make it into English in the early 1800s:
Mishcàt-Ul-Másàbih or collection of the most authentic traditions regarding the actions and sayings of Muhammed, Vol 2, by Muḥammad Ibn-ʻAbdallāh al- H̲aṭīb at-Tibrīzī (Arnold N. Matthews, translator), Hindoostanee Press, 1810;
The Pictorial Bible, Being the Old and New Testaments, According to the Authorized Version, Vol 1 published by Charles Knight, 1836; commentary not credited.
2. On the function, utility, and fragility of the nose – Early modern patients and their surgeons, Mariacarla Gadebusch Bondio, Nuncius: Journal of the Material and Visual History of Science 32 (2017), 25-52.
3. Well, traditionally, it is said to have been an alloy of silver and gold; a recent analysis of the staining around his sinuses indicates that it may actually have been copper and zinc. Perhaps he had more than one, and a servant decided that the corpse could do with second best. Source.
4. Otolaryngology: An illustrated history by N. Weir, published by Butterworths, 1990. (Sadly, this fact is not among those illustrated).
5. Despite the name, the Tin-Noses Shop that provided reconstructive masks for soldiers disfigured in WWI used galvanized copper covered by enamel paint, not tin. These replaced earlier rubber masks that proved insufficiently durable for daily use. Tin really seems not to be a popular nose material, Cat Ballou notwithstanding.