Martial wrote a poem about Publius' dog called Issa. It begins:
Issa est passere nequior Catulli,
Issa est purior osculo columbae,
Issa est blandior omnibus puellis,
Issa est carior Indicis lapillis,
Issa est deliciae catella Publi.
Issa is naughtier than Catullus’ sparrow,
Issa is purer than a
dove’s kiss,
Issa is more winning than any girl,
Issa is more
precious than Indian pearls,
Issa is a lapdog, Publius’ darling.
Epigrams, 1.109.1-5 (trans. D. R. Shackleton-Bailey)
The whole poem is uncharacteristically sweet. If it's actually full of euphemisms for sordidness, like so much of Martial, please don't tell me.
Also, Ovid names some of Actaeon's dogs in the Metamorphoses, 3.206-225:
Dum dubitat, videre canes, primique Melampus
Ichnobatesque sagax latratu signa dedere,
Cnosius Ichnobates, Spartana gente Melampus.
inde ruunt alii rapida velocius aura,
Pamphagos et Dorceus et Oribasos, Arcades omnes,
Nebrophonosque valens et trux cum Laelape Theron
et pedibus Pterelas et naribus utilis Agre
Hylaeusque ferox nuper percussus ab apro
deque lupo concepta Nape pecudesque secuta
Poemenis et natis comitata Harpyia duobus
et substricta gerens Sicyonius ilia Ladon
et Dromas et Canache Sticteque et Tigris et Alce
et niveis Leucon et villis Asbolos atris
praevalidusque Lacon et cursu fortis Aello
et Thoos et Cyprio velox cum fratre Lycisce
et nigram medio frontem distinctus ab albo
Harpalos et Melaneus hirsutaque corpore Lachne
et patre Dictaeo, sed matre Laconide nati
Labros et Argiodus et acutae vocis Hylactor
But while he stands perplexed he sees his hounds. And first come
Melampus and keen-scented Ichnobates, baying loud on the
trail—Ichnobates a Cretan dog, Melampus a Spartan; then others come
rushing on swifter than the wind: Pamphagus, Dorceus, and Oribasos,
Arcadians all; staunch Nebrophonos, fierce Theron and Laelaps;
Pterelas, the swift of foot, and keen-scented Agre; savage Hylaeus,
but lately ripped up by a boar; the wolf-dog Nape and the trusty
shepherd Poemenis; Harpyia with her two pups; Sicyonian Ladon, thin in
the flanks; Dromas, Canache, Sticte, Tigris, Alce; white-haired
Leucon, black Asbolos; Lacon, renowned for strength, and fleet Aëllo;
Thoos and swift Lycisce with her brother Cyprius; Harpalos, with a
white spot in the middle of his black forehead; Melaneus and shaggy
Lachne; two dogs from a Cretan father and a Spartan mother, Labros and
Argiodus; shrill-tongued Hylactor ...
The English translations of these would be, in order (NB: revised since first posting):
Black-foot, Trail-follower, Voracious, Gazelle, Mountain-ranger, Faun-killer, Hunter (Theron from θήρα, hunting), Hurricane, Winged, Hunter ("keen-scented" Agre), Sylvan, Glen, Shepherd, Seizer, Catcher, Runner, Gnasher (or Noisy from καναχός), Spot, Tigress, Might, White, Soot, Spartan, Whirlwind, Swift (Thoos from θοός, quick, nimble), Lycisce (perhaps related to λύκος, wolf), Cyprian, Grasper, Black, Shaggy, Fury (Labros from λάβρος, furious), White-tooth (Argiodus a compound of ἀργής, white + ὀδούς, tooth), Barker (Hylactor from ὑλακτέω, to bark or howl). (List adapted from trans. Frank Justus Miller, rev. G. P. Goold)
These seem to suggest that a pattern for naming dogs was simply to call them by their attributes. Xenophon advises giving hunting dogs short names, "so as to be able to call to them easily". He continues:
εἶναι δὲ χρὴ τοιάδε, Ψυχή, Θυμός, Πόρπαξ, Στύραξ, Λόγχη, Λόχος,
Φρουρά, Φύλαξ, Τάξις, Ξίφων, Φόναξ, Φλέγων, Ἀλκή, Τεύχων, Ὑλεύς,
Μήδας, Πόρθων, Σπέρχων, Ὀργή, Βρέμων, Ὕβρις, Θάλλων, Ῥώμη, Ἀνθεύς,
Ἥβα, Γηθεύς, Χαρά, Λεύσσων, Αὐγώ, Πολύς, Βία, Στίχων, Σπουδή, Βρύας,
Οἰνάς, Στερρός, Κραύγη, Καίνων, Τύρβας, Σθένων, Αἰθήρ, Ἀκτίς, Αἰχμή,
Νόης, Γνώμη, Στίβων, Ὁρμή.
The following are the right sort: Psyche, Thymus, Porpax, Styrax,
Lonché, Lochus, Phrura, Phylax, Taxis, Xiphon, Phonax, Phlegon, Alcé,
Teuchon, Hyleus, Medas, Porthon, Sperchon, Orgé, Bremon, Hybris,
Thallon, Rhomé, Antheus, Hebe, Getheus, Chara, Leusson, Augo, Polys,
Bia, Stichon, Spudé, Bryas, Oenas, Sterrus, Craugé, Caenon, Tyrbas,
Sthenon, Aether, Actis, Aechmé, Noës, Gnomé, Stibon, Elormé.
On Hunting, 7.5
The translators (Marchant & Bowersock) note: "The names are significant of the colour, strength, spirit, sagacity or behaviour of the hounds. Hebe and Psyche are still in the list of bitches’ names, and modern equivalents of several of the other names are in use, e.g. Lance (Lonché), Sentinel (Phylax), Ecstasy (Chara), Blueskin (Oenas), Crafty (Medas), Hasty (Sperchon), Vigorous (Thallon), Impetus (Hormé), Counsellor (Noës), Bustler (dog) or Hasty (bitch); cf. Sperchon. For Πολύς we should probably read Πολεύς, “Rover.”"