I'm curious whether any canonical explanation has ever been given. It seems that Time Turners must be extremely difficult to create, seeing that it's mentioned in the books that their use is tightly controlled by the Ministry of Magic (so obviously a wizard can't just wave their wand over an hourglass and turn it into a Time Turner); also because they mess with the very fabric of reality (so it's hard to imagine there being some naturally occurring magical material that just happens to be able to enable time travel.)
Pottermore has some discussion of the Ministry research into time travel. Here’s an excerpt (emphasis mine):
According to Professor Saul Croaker, who has spent his entire career in the Department of Mysteries studying time-magic:
“As our investigations currently stand, the longest period that may be relived without the possibility of serious harm to the traveller or to time itself is around five hours. We have been able to encase single Hour-Reversal Charms, which are unstable and benefit from containment, in small, enchanted hour-glasses that may be worn around a witch or wizard’s neck and revolved according to the number of hours the user wishes to relive.”
That sounds like a Time-Turner, doesn’t it? This confirms your suspicion that Time-Turners are manufactured, and don’t just spring into being. It also gives the best explanation I can find of how they’re made.
The idea that the Ministry is the sole proprietor of Time-Turners (at least in the UK) is confirmed in other parts of the canon. In Order of the Phoenix, we visit the Dept. of Mysteries and a cabinet full of Time-Turners:
The jet of red light had flown right over the Death Eater’s shoulder and hit a glass-fronted cabinet on the wall full of variously shaped hour-glasses; the cabinet fell to the floor and burst apart, glass flying everywhere, sprang back up on to the wall, fully mended, then fell down again, and shattered—
while the glass-fronted cabinet that Harry now suspected had contained Time-Turners continued to fall, shatter and repair itself on the wall behind them.
JKR said in a 2007 interview that this battle the Ministry shattered “all the Time-Turners”, and meant Harry and co. couldn’t use any, so the Ministry is the sole source of Time-Turners.
Their research goes further: there’s a bell jar that contains (as thegoose puts it) “liquid time”. There’s a scene in the battle where a Death Eater gets his head stuck in the jar and ends up with a baby’s head on an adult’s body. Alternatively, here’s a description from when they first meet it:
He led the way down the narrow space between the lines of desks, heading, as he had done in his dream, for the source of the light, the crystal bell jar quite as tall as he was that stood on a desk and appeared to be full of a billowing, glittering wind.
“Oh, look!” said Ginny, as they drew nearer, pointing at the very heart of the bell jar.
Drifting along in the sparkling current inside was a tiny, jewel-bright egg. As it rose in the jar, it cracked open and a hummingbird emerged, which was carried to the very top of the jar, but as it fell on the draught its feathers became bedraggled and damp again, and by the time it had been borne back to the bottom of the jar it had been enclosed once more in its egg.
This seems like very advanced magic, probably requiring years of specialist study. This jar goes well beyond single hour-reversal, as it reverses part of the Death Eater’s life, a period of many years. Unfortunately I don’t think we get a more detailed explanation of what’s in this jar, or how it works.
In Pottermore (if anyone has images that'd be great) it is explained that Time-Turners are created as an experiment by the Ministry of Magic - and more specifically, the Department of Mysteries.
This is affirmed in-universe during the untitled Battle of the Department of Mysteries, where Harry, Ron and Hermione (along with members of the DA) duel Voldemort's Death Eaters.
During that battle, the cabinet holding the Time Turners is smashed - located outside of the Hall of Prophecies, and near the liquid Time and the Veil.