OK, to follow up on my earlier question about the earliest named cat…
Do we have any info on when humans started naming birds. Not, like, species. I want to know when we started naming birds individually?
Was there an old Greek guy with a bird called ‘pretty bird’ or ‘polly’ or whatever the equivalent was in Greek?
Do Huginn and Muninn count?
Catullus (1st century BCE) wrote three poems about his lover’s pet bird, how she loved it and how she mourned when it died. I don’t think the bird was ever named in the poems, but I can’t believe it didn’t have a name.
Emperor Honorius had a hen named Roma which it was said he loved more than Rome itself.
I’m no scholar of ancient history, but I do have access to a number of history and archaeological journals through work, and a cursory search suggests that, at least in the Western world, it is highly unusual for animals to be named in writing, even in works (like Catullus’s poems about his lover’s “sparrow”, or Pliny’s story about the cobbler’s raven) that are specifically about individual animals. This suggests either that ancients tended not to name their pets, or at least that they didn’t attach much importance to the names. Of course, there are some exceptions that prove the rule. Aelian wrote about an elephant named Nikaia; the fact that Aelian references her by name is remarkable to modern scholars (“The exceptional role of the female elephant is further emphasized by the fact that, in contrast to many other animals that occur in ancient literature, she has a name…” writes Thorsten Fögen in a 2007 article “Pliny the Elder’s Animals”). Another article I found, “Greek and Roman Household Pets”, mentions that epitaphs for pets were fashionable in the ancient world, and quotes one for a pet nightingale. But I don’t know enough Ancient Greek to tell whether it refers to the bird by name. The epitaph appears on the front page of the article, which is probably the only page that isn’t paywalled, so maybe some Doper with a classical education can have a gander. 
Circa 330 BCE, Alexander the Great named his horse “Bucephalas”. It would seem odd to me if his contemporaries did not name their pets.
He also had a greyhound named Peritas, and both animals had cities named after them and monuments raised in their honour, so I suspect he may have gone quite a bit beyond what was normal in that regard. Although, I guess it is also a large part of the reason we know anything at all about them today, so points to Alexander I suppose.
It seems equally odd that Pliny the Elder, a historian revered for his attention to detail, dedicated nearly an entire chapter of his encyclopedia to a beloved pet raven, the subject of a public funeral more elaborate than that accorded even to famous generals, without bothering to mention its name.