Medievalist has an article on Medieval Pet names:
In England we find dogs that were named Sturdy, Whitefoot, Hardy, Jakke, Bo and Terri.Anne Boleyn, one of the wives of King Henry VIII, had a dog named Purkoy, who got its name from the French ‘pourquoi’ because it was very inquisitive.
Geoffrey Chaucer’s The Nun’s Priest Tale has a line where they name three dogs: Colle, Talbot and Gerland. Meanwhile, in the early fifteenth-century, Edward, Duke of York, wrote The Master of Game, which explains how dogs are to be used in hunting and taken care of. He also included a list of 1100 names that he thought would be appropriate for hunting dogs. They include Troy, Nosewise, Amiable, Nameles, Clenche, Bragge, Ringwood and Holdfast.
Meanwhile, in Switzerland a list of 80 dogs that took part in a shooting festival in the year 1504 has been preserved. They reveal the most popular name was Furst (Prince). Other names included Venus, Fortuna, and Turgk. Some dogs got their names from the work being done by their owners: Hemmerli (Little Hammer) belonged to a locksmith, while Speichli (Little Spoke) belonged to a wagoner.the article also notes the names of medieval cats.
Ancient Egyptians also named their pets (and the article includes many pictures of dogs)
We even know many ancient Egyptian dog's names from leather collars as well as stelae and reliefs. They included names such as Brave One, Reliable, Good Herdsman, North-Wind, Antelope and even "Useless". Other names come from the dogs color, such as Blacky, while still other dogs were given numbers for names, such as "the Fifth".
Read more: http://www.touregypt.net/featurestories/dogs.htm#ixzz2XBoSftFj
and of course, the most famous dog of Antiquity is Ulysses' dog Argos,
(in Greek: Ἀργώ, meaning 'swift')