The simple truth is that most of the brighter stars that we see have common names. This means it would have been very hard to name them in an order of decreasing brightness if there was no naming system like the Bayer Naming System. With this systematic method, all the brighter stars in the constellation now have their specific names.
The Bayer Naming System assigns a name to each brighter star by using a Greek letter such as "Alpha, Beta, Gamma," and a Latin word that describes the common name of the star. For instance, while a common name for a star is Orion, the Bayer Naming System gives specific names such as Alpha Orionis, Beta Orionis, etc.
Having known the naming system used by Johann Bayer for stars in the constellation, the name given to the brightest star is Sirius. However, in the Bayer Naming System, Sirius is also known as Alpha Canis Majoris.