The only clues are (1) we’re told that he’s a hunting dog; (2) in terms of story structure, he plays the role of a watchdog; and (3) there could in principle be a clue in the name Argos.
The fact that he’s a hunting dog gives us some pointers. But bear in mind that dog breeds change very quickly: a given breed from just one century ago can be startlingly different from a dog of the same breed today.
The fact that he’s a hunting dog points to one of the two known breeds of hunting dogs in Greek antiquity, namely the Lakonian and Molossian. Xenophon’s work On hunting (with dogs) (Kynegetikos) recommends two varieties of Lakonian, which he calls the ‘Castorian’ (kastoria), supposedly linked to the mythological figure Kastor, and ‘Vulpine’ (alōpekis) because it’s supposedly a hybrid between the dog and the fox (Kynegetikos 3.1). He seems to prefer the Castorian: his ideal physical characteristics are (4.1-2, tr. Marchant)
First, then, they should be big. Next, the head should be light, flat, and muscular; the lower parts of the forehead sinewy; the eyes prominent, black, and sparkling; the forehead broad, with a deep dividing line; the ears small and thin with little hair behind; the neck long, loose, and round; the chest broad and fairly fleshy; the shoulder-blades slightly outstanding from the shoulders; the forelegs short, straight, round, and firm; the elbows straight [i.e. not bent inwards or outwards]; the ribs not low down on the ground [i.e. when the dog is couchant], but sloping in an oblique line; the loins fleshy, of medium length, and neither too loose nor too hard; the flanks of medium size; the hips round and fleshy at the back, not close at the top, and smooth on the inside; the under part of the belly and the belly itself slim; the tail long, straight, and thin; the thighs hard; the shanks [elbows to feet] long, round, and solid; the hind-legs much longer than the forelegs and slightly bent; the feet round. Hounds like these will be strong in appearance, agile, well-proportioned, and speedy; and they will have a jaunty expression and a good mouth.
Even this is a good 250 years later than the Odyssey, so it’s likely that breeds changed significantly in the intervening centuries; and there’s obviously no guarantee that Homer liked the same kinds of dogs that Xenophon did. But it’s the closest benchmark we’ve got.
As for Argos’ name, there isn’t much to go on there. Árgos is believed to come from an adjective argós, with a regular shift in accent. Argos can mean ‘fast’ and ‘bright/white’: it’s been debated whether it was originally two separate words, but they are probably the same root, since cognates appear with both meanings in Sanskrit (r̥jrá-, r̥ji-).
The meaning ‘fast’ appears in Homer when argos is used as an independent adjective, e.g. kynes podas argoi ‘dogs swift of foot’.
The meaning ‘bright, white’ appears in Homer in several related words: argēs (used of lightning, animal fat, and cloth), argennos (used of sheep), enargēs (of a dream; gods are not enargēs to see; conventionally translated ‘plain to sight’), argikeraunos (of Zeus: ‘of the bright thunderbolt’), and argiodous (argos-toothed, of boars and dogs). It’s also related to the word for ‘silver’, argyros.
‘Argos’ is the only dog name that appears in early Greek epic, so it’s hard to evaluate its significance beyond these points. It’s possible that it’s intended as a typical dog name; we can’t be certain. Argos’ name is homophonous with the name of a city, but they’re unrelated: Argos the city is a neuter e-stem noun and probably pre-Greek, Argos the dog is a masculine o-stem noun with Indo-European origins.
Argos the dog does however share his name with the creature placed as a guard on Io. Io and Argos aren’t mentioned in Homer; their earliest attestation is in Aischylos. Still, it’s quite possible that the parallel indicates that Argos is a typical watchdog name.