Of course they named their animals! The Roman poet Ovid re-tells the story of Diana and Actaeon in his epic Metamorphoses (III.206-252), in which the hunter Actaeon accidentally gets a peep at Diana while she is bathing. Mortals can't see a goddess nekked without consequences, and so Diana makes Actaeon's own hunting dogs tear him apart (not fair, I know). Ovid gives us a wonderful list of names for the dogs:
Aello (Ἀελλώ) – “Tempest”
Agre (Ἄγρα) – “Chase”
Alce (Ἀλκή) – “Stout”
Argiodus (---) – “White-Tooth”
Asbolos (Ἀσβόλος) – “Soot”
Canache (Καναχή) – “Noisy”
Cyprius (---) – “Cyprian”
Dorceus (Δορκεύς) – “Gazelle”
Dromas (Δρομάς) – “Runner”
Harpalos (< ἁρπάζω) – “Grasper”
Harpyia (Ἅρπυια) – “Snatcher”
Hylactor (< ὑλακτέω) – “Howler”
Hylaeus (Ὑλαῖος) – “Woody”
Ichnobates (Ἰχνοβάτης) – “Trail-Seeker”
Labros (Λάβρος) – “Fierce”
Lachne (Λάχνη) – “Shaggy”
Lacon (Λάκων) – “Spartan”
Ladon (Λάδων) – “Catcher”
Laelaps (---) – “Hunter”
Leukon (Λευκόν) – “White”
Lycisce (Λυκίσκος) – “Wolf-Dog”
Melampos (Μελάμπους) – “Black-Foot”
Melanchaetes (Μελαγχαίτης) – “Black-Hair”
Melaneus (Μελανεύς) – “Black”
Nape (Νάπη) – “Vale”
Nebrophonos (Νεβροφόνος) – “Fawn-Killer”
Oresitrophos (---) – “Crag-Born”
Oribasos (Ὀρείβασος) – “Mountain-Climber”
Pamphagos (Παμφάγος) - “Everything-Eater”
Poemenis (Ποιμήν) – “Shepherd”
Pterelas (Πτερέλας) – “Winged”
Sticte (Στικτή) – “Spot”
Theridamas (---) – “Killer”
Thēron (---) – “Whirlwind” [or “Summer”]
Thoos (Θοός) – “Quick”
Tigris (Τίγρις) – “Tiger”
Greetings! I asked a similar question, specifically about ancient Egypt back in Oct. 2015. I don't have any information myself, but I'll link you to my thread, which has a few magnificent answers.
Also in Homer's Odyssey, Odysseus had a dog called Argos (Αργος).