In the truest oldest stories of Greek mythology, the owl associated with Athena bore no name, nor was it strictly speaking "her owl", but rather one of the manifestations of Athena herself.
"The significance of Athena's owl culturally/symbolically in ancient times is that it is not Athena's owl but Athena = owl. Most of the images of an owl in ancient Greece simply represent Athena. Only rarely does she appear with a symbol of an owl and she never seems to appear with an owl. She was involved with shape shifting and she would sometimes appear as an owl. But the owl represents wisdom."
One word in Greek that meant owl was Glaukos . It is also a a proper name. It is unlikely that the owl had a name, but if it did, this would certainly have been a possibility.
There is a lot of speculation about why the owl came to be associated with Athene and wisdom. Some theorize that the way that the owl's eyes glow reflected the inner light of wisdom caused the ancient Greeks to associate it with Athene. Others speculate that one particular species of owl-
Athene noctua- or the Little owl- was prevelent in the area and made it's nest in the Acropolis itself.
" Even the latin name, translates to “Athena’s Night” or “Athena of the Night”. At about 9″ it is perhaps slightly larger than a Northern Saw-Whet Owl. It is common across Europe, North Africa the Middle East in Asia. It also has a habit of nesting in buildings. No doubt the Parthenon on the Acropolis in Athens would have been very attractive to this species."
Now, there was a movie, called Clash of the Titan's that came out in 1981. In this movie, Athene's owl did indeed have a name. It was called Bubo. In the movie Athene is told to give Perseus her owl...
"Your friend--Bubo the Owl. Give Perseus your friend. It is all knowing, all seeing. Give it to him." He continues, in a boastful manner: "It is my wish. My ... command."
Now the name Bubo- is less mythologically significant than it is scientifically. It is the name of a genus of owls.
"The call of the Eagle Owl is a deep resonant “ooh-hu” with emphasis on the first syllable for the male, and a more high-pitched uh-Hu for the female (in German, the name of this bird is "Uhu"). A recent study has shown that these calls are easily individualized. It means that each member of an Eagle Owl population can be individually identified by means of its vocalizations  . The Eagle Owl's scientific name comes from its deep booming call. The word "bubo," based on the call, has referred to owls, particularly the Eagle Owl, since the Dark Ages, and is mentioned in Middle English bestiaries."
Hope this helps!