Despite what some companies claim, it is not possible to buy the naming rights to a star. All the stars that can be seen in the night sky either have been named long ago or have various catalog designations.
Stars with Multiple Designations
Many stars have more than one name. For example, Sirius, the brightest star in the sky, is also known as Alpha Canis Majoris, Canincula, the Dog Star, Aschere, SAO 151881, GSC 5949:2777, HD 48915, HR 2491, and B-16 1591. Its most common name, Sirius, means "scorching" in Greek, which is a fitting name for the brightest star in the sky. The nickname the Dog Star comes from the star being in the constellation Canis Major, the Great Dog. And its designation of Alpha Canis Majoris signals it as the brightest star in the constellation Canis Major. (However, the brightest star in a constellation is not always the Alpha star, see below.) Most of the other identifiers are numbers depending on which catalog is being referred to. These catalog numbers are also how any newly found stars are named.
Greek and Arabic Star Names
Most of the stars in the sky are given Greek and/or Arabic names. Johann Bayer, in 1603, completed maps known as Uranometria in which he designed Greek letters as names for the constellations' brightest stars. However, he did not hold fast to designating the brightest Alpha, the next brightest Beta, and so on. Sometimes he started at one end and simply lettered them until he got to the other end, other times he lettered within a cluster and then moved outward in the constellation, among various other methods. Bayer's decision on the importance of a star was more likely to influence his Alpha designation.
Besides Greek letters, the stars are then paired with the Latin genitive of their constellation. Therefore we have Alpha Canis Majoris and not Alpha Canis Major, or Beta Geminorum and not Beta Gemini. When written on maps, the lower case Greek letter is always used. Double stars can also be given the same designation except with a number added, such as Alpha 1 and Alpha 2 Librae.
The Arabic names given to many of the stars have been changed slightly over time, through variant spellings, pronunciations, and translations. The Arabic names often indicate the specific part of the constellation that the star is supposed to represent, such as "the forehead" (Algieba), "the thigh" (Phecda), and "naval of the steed" (Alpheratz).
Other than Greek and Arabic names and catalog numbers, there are other names that have sneaked in over time. One great example of this is the stars named Sualocin and Rotanev in Delphinus. These stars were most likely named by an observatory assistant named Nicolaus Venator, who hid his own name by spelling it backward and then placing it on star maps.
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