It seems that all stars as bright or brighter than Polaris have names dating to ancient times, including many stars dimmer than Polaris. So what was Polaris called, in any language, before it became primarily known for its proximity to the celestial pole?
Polaris is a very well known star and named in many different languages. For example, in Inuit astronomy it was known as Niquirtsuituq and is depicted in the flag and coat of arms of the Canadian Inuit territory of Nunavut as well on the flag of Alaska. In fact, the Inuits called the three stars, Polaris, Kochab & Pherkad, Nuutuitut, the never-moving.
It is part of the constellation Ursa Minor. Since constellations are fixed despite precession it would have been first known as belonging to that constellation. In fact, at first the entire constellation was used for navigation.
Polaris is actually a triple star system with it's most prominant star being a yellow gas giant. Planets have been detected around four of the stars in the constellation and the constellation also includes an isolated neutron star, Calvera, as well as the hottest white dwarf star ever found with a surface temperature of 200,000 K.
In Europe, it was named Polaris in the Renaissance from the Latin Stella Polaris or pole star when it approached tge celestial pole to within a few degrees. In Antiquity it was not close enough to the celestial pole to be used for navigaton and the whole constellation was used. In Old English it was called scip steorra, or ship star.
In the Babylonian star catalogues Ursa Minor was known as the Wagon of Heaven, Mul.mar.gid.da.an.na associated with the goddess Damkina. It's named in the Mul.apin catalogue whose earliest recorded copy is dated to 686 BCE but thought to have been first compiled around 1000 BCE.
According to Diogenes Laertius and citing Callimachus, Thales of Miletus 'measured the stars of the wagon by which the Phoenicians sail'. Diogenes identified this with the constellation Ursa Minor. The Greeks also reported that the Phoenicians used this star for navigation at sea and for this reason called it also Phoineke. The tradition of naming the constellation as a bear appears to be genuinely Greek although Homer speaks of only one bear.
The original bear is thus Ursa Major, and Ursa Minor was admitted as a second or Phoenician Bear (Ursa Phoenica) only later, and according to Strabo, this was due to a suggestion by Thales that it be used as a navigation aid by the Greeks who had been using Ursa Major in this way. Given the above, this is unlikely to have been an independent discovery by Thales, despite how Strabo reported it, but he was simply reporting on how the Phoenicians were already using this constellation in navigation.
Except for very few conspicuous stars (like Sirius), most stars had no individual names in the ancient Greece and Rome. In particular, in Ptolemy catalog, no star in Ursa Minor has a name. Polaris is described in this catalog as "the star at the end of the tail".
Same applies to the earliest surviving Greek text on astronomy: the poem Phenomena of Aratus.
It was in the medieval time when stars were given individual names. And "stella polaris" obtained this name as a conspicuous star close to the pole.
Remark. Aratus gives an alternative name for the Little Bear: Κυνόσουραν, and mentions that it is used for navigation since it is close to the pole, but does not mention any individual stars in it.
I have a copy of Richard Hinckley Allen's Star Names: Their Lore and Meaning,1899,1963.
The section on Polaris, Alpha Ursae Minoris, extends from page 453 to page 458.
It says the early Greek name for Polaris was Phoenice.
Pages 456 to 457 give its names in China (Pih Keih and others), in India (Grahadara or Dhruva), among the Arabs (Al Kiblah and Al Kaukab al Shamalyy), & the Turks (Yilduz).
This book should definately be consulted by anyone interested in the names of stars.
Stars of Jade: Astronomy and Star Lore of Very Ancient Imperial China Julius H.W. Staal, 1984, has a lot of information about Chinese constellations and star names.
And I suspect that there are other sources for star names in many other cultures.
According to Wikipedia, the Inuit call Polaris Niqirtsuitug.
But I don't know when those various names were adopted. As the north celestial pole got closer and closer to Polaris, it became more likely that a particular culture would give the star a name referring to its position on the celestial sphere.
In some cases those "polar" names might have been the first names given to Polaris by a particular culture. But in some cultures it might have been the practice to name almost every star visible to the naked eye, presumably by their position with an asterism or constellation, and many cultures used asterisms and constellations different from modern ones. Thus Polaris might have had names in some cultures which referred to it being, for example, at the end of the handle or tail of an asterism or constellation, before that culture named it the pole star.
Since Polaris is very close - about 0.75 of a degree - to the celestial north pole, it is only visible in the Northern hemisphere - and a tiny distance south of the equator at some times.
So the native cultures of Australia, southern Africa, and most of South America would never have had any names for Polaris until people from the northern hemisphere arrived.
As al-Jadi, Polaris was part of a Bedouin or Arab legend of the sky, which told of a funeral for a father. Polaris was the murderer. It's important, if the meaning is 'goat', which I'm not sure yet but seems pretty likely, various star names today take an Arab name and vary the spelling for several stars. Alpha Capricornus is alGedi, sometimes spelled Giedi. Arab/ Bedouin astrology had its own animals and constellations and asterisms. There are also multiple instances of various animals. I have a file listing them all as I find them. For example, there are so many camel stars, with variations such as mother camel (4 of them), baby camel, trotting camel and so on. Also just because goat appears as Capricorn to us doesn't mean it was a goat in Arab astrology (I couldn't say presently) or that there weren't other goats. But in this case it's a human story, the stars in the Big Dipper feature, the handle of the dipper is the mourner sisters, mourning there father in funeral procession, the cup of the ladle is the coffin which they are following.
I myself defer to the Arabic traditions for various reasons, personally I dislike hybrid monster type depictions such as goat-fish of Capricorn. Then also Arab/ Bedouin astrology is clear-headed clear-skied astrology/ astronomy from a culture that used the stars for navigating ON LAND. Their feet are very much on the ground, and they are able to see the stars at most times due to lack of cloud cover. Although at some time in the distant past the region wasn't desert, but I think that is going back way far, beyond a full cycle of the pole stars.
I have also been searching a name for Polaris. The Babylonian name I saw here refers to constellation so still not the wiser exactly. But the dates for that are surprisingly recent. Polaris became pole star only in 500 AD, so it can still be relevant. I'm glad others are trying to answer this question, as I have been trying now and then for a while. I had high hopes for Phoenice, but now I see it's also a pole star name. I've been through most of the other names listed here but haven't found one better than al-Jadi yet. Which name may relate to the pole star 'murdering' the other previous pole star. Which there wasn't a good one between Thuban and Polaris but Kochab was used as a stop-gap.
I checked out some names and found them to refer to an axle (al-Kutb) and a needle or nail (Mismar). I guess that such names also relate to Polaris being at the center of the revolving stars. Moving forward I think al-Jadi is good enough for future use, and has the advantage of carrying the full Arabic myth of the funeral procession. Looking back on our generation, people would associate the name with our generations having developed a lot of science and seeing our own era as pivotal, a new era, out with the old and death to the old (al-Jadi as legendary murderer). I'd love to know more about the myth, as in who was murdered exactly and why. al-Na'ash is what I have. But that refers to bier, coffin, casket, so seems is not a name.
Some names for Polaris just mean star, such as Yilduz from Turkish, the same word appearing in form Yildun for a different star. (Similarly it seems Kochab the former pole star is from kaukab in Arabic meaning star.) So again meaning pole star so THE star.
I will try to link a graphic here using the instructions given. It just shows my efforts to clarify the whole cycle of pole stars. Emphasizing why we want a genuine star name, as any one of these stars could be 'Polaris' in their own time, given it is a wide time frame. !https://www.facebook.com/PracticalHeroesAstrology/photos/a.1887269767966380/5467568069936514/ [Pole Stars]