Bright stars from first to third magnitude have proper names that have been in use for hundreds of years. Most of them are Arabic (e.g., Betelgeuse, Dubhe). J. Bayer of Bavaria introduced his system for star names in 1603. Bayer assigned successive letters of the Greek alphabet to the brighter stars of each constellation. Each Bayer designation is the Greek letter with the genitive form of the constellation name.(Polaris is Alpha Ursae Minoris. (Although he wasn't consistent: Bayer switched brightness order for serial order in assigning Greek letters in the Ursa Major, going from cup to handle).

Faint stars are designated in catalogs like the The Smithsonian Astrophysical Observatory (SAO) Catalogue, the Yale Star Catalog, and The Henry Draper Catalog, and the Bonner Durchmusterung, compiled at Bonn Observatory starting in 1837. A third of a million stars are listed by "BD numbers." These procedures and catalogs accepted by the International Astronomical Union are the only means by which stars receive long-lasting names. So if you're buying a name from the Celestial Registry, you're doing so for your own amusement

Source: International Planetarium Society