The brightest stars in the sky all have names that were given by ancient civilizations. Most of the names we use today have been passed down from Greek and Roman cultures. These include star names such as Beteleguese (in Orion), Sirius (in Canis Major), Arcturus (in Bootes), Spica (in Virgo), and many others.
Another way of naming stars is to assign a letter from the Greek Alphabet (Alpha as the brightest) along with the name of the constellation they are in. For instance the brightest star in Leo would be Alpha Leonis. The second brightest would be Beta Leonis, etc.
One thing to know is that the only organization that has the official rights to naming stars is the International Astronomical Union. It is simply not possible to have a star named after a person. You may pay $50 to some group for a fancy certificate, but no one will ever refer to that star as "Edna Garffdangle".