How many stars can you name? And do you think a professional astronomer would do better or about the same?
Too many to remember
Everyone can name at least one star - the name of Earth's daytime star in your own language! It gets harder after the Sun sets and the dark sky reveals Sirius, Betelgeuse, Polaris, Arcturus or Vega. These are the common names of some bright stars, but with the unaided eye you might see a few thousand stars from a dark location.
With a telescope it gets even more difficult, especially if we consider large telescopes and space telescopes. The Hubble Space Telescope Guide Star Catalog lists 19 million dim stars. There wouldn't even be time to give them all unique names, let alone learn them.
As it happens, astronomers don't name stars. In addition, most professionals rarely use star names as it isn't useful for their work. That means you may well know just as many star names as the pros, perhaps more.
Stars and constellations
There are a few hundred individual star names, almost all coming to us from the Arabs and the Greeks of many centuries ago. They named bright stars or members of prominent constellations and asterisms. (An asterism is a group of stars which is part of one or more constellations.)
In fact it's the groups of stars that have always been the key to knowing the sky. Even a star's name often refers to its place in a constellation. For example, Rigel in Orion is the hunter's left foot. The name just means foot. The star Alpha Centauri is also known as Rigil Kentaurus, the Centaur's foot.
Constellations have changed over millennia and from culture to culture, but are a feature of astronomy throughout history. People could learn to recognize groups of stars, and let the sky serve as a clock or calendar or navigation aid. Learning a thousand star names would be like memorizing a thousand phone numbers.
Bayer and Flamsteed
If giving each star a name isn't practical, then you need another system to keep track of them. In 1603 German astronomer Johann Bayer (1572-1625) published a star atlas using Greek letters for the stars in a constellation, usually in order of brightness. This is how Rigil Kentaurus came to be known as Alpha Centauri. You'll notice that the form of the constellation name isn't the familiar one. Bayer used the Latin possessive form of the name, so Alpha Centauri means "alpha of Centaurus".
With the invention of the telescope they had to account for many more stars. Since constellations have more stars than the Greek alphabet does letters, the Bayer designations are limited. But as you can see in this star map of southern constellations, they're still in use.
Another way of identifying stars is by using catalog numbers. The catalogs have changed, but not the idea. The star atlas of John Flamsteed (1646-1719) was the first major one to be based on telescopic observations and it was the most authoritative one available for many decades. The numbers however were added to the catalog later by French astronomer Joseph Jerome de Lalande (1782-1807).
Flamsteed numbers are used for stars without a Bayer designation. They follow the same pattern, but numbers replace Greek letters. A famous example is 51 Pegasi, which is the first Sun-like star found to have a planet orbiting it.
Henry Draper and beyond
Often we see stars identified by an HD number. Betelgeuse, for example, is HD 39801, but stars without distinct names are the ones most likely to be shown in this way. "HD" refers to the Henry Draper catalog. A number of star catalogs bear the name of the compiler, but not this one. Although Draper planned a catalog with stars classified by their spectra, he died before his project really got going.
Henry Draper's widow gave a sum of money to Harvard College Observatory to carry out the project. The catalog contains spectra of over a quarter of a million stars, most of the work done by Annie Cannon.
The large sky survey projects of today generate catalogs by computer, with the stars identified by their positions in the sky at a particular time.
What the IAU does
The International Astronomical Union (IAU) is in charge of standards for naming heavenly bodies. The naming and monitoring of minor planets requires considerable oversight, but since stars aren't named there isn't much to do in that regard. The IAU did however tidy up the constellations. In 1922 the entire sky was divided up into 88 clearly-defined constellations.
With regard to star names the IAU does maintain a database of catalogs. Each new one gets a unique designation of initial letters, to ensure that the stars are identified unambiguously.
Can you name a star?
The simple answer is: Yes. You can name a star anything you like, make a certificate for yourself, put it on your Facebook page and tweet it to all your followers. And, at no cost, your star name will have as much official recognition as the ones you can buy from commercial companies, i.e., none.
It's not illegal to name stars or to charge for doing it. However the seller mustn't say (or imply) that any name will be officially recognized, because the IAU does not sanction this.
What the IAU says about buying star names