Actually, you can choose any name you want.
It can be a real name, such as "John Miller",
a nickname, such as "My Sweetheart"
a general term, such as "Our Family Star"
or (referring to the binary stars) two names, such as "Tina and John".
It's your choice. The only limit you got, is the limit of 35 characters, which must not be exceeded, to keep the format of the star naming certificate.
(In exceptional cases we reserve the right to not tolerate a star name. For example political statements or insults.)