Odysseus is the full Greek name of the hero known in Latin as Ulysses — the wandering, cunning protagonist of Homer's Odyssey. For a male dog, it's one of the most overtly literary pet name choices possible: three syllables of ancient Greek epic weight, chosen by someone with deep affection for the source material.
The Classical Mythology Angle
Odysseus sits at the extreme end of the classical mythology naming tradition — further out than Zeus or Apollo, which have crossed into the mainstream, and well past Hermes or Achilles. Odysseus is a specific literary reference rather than a general mythology reference: you're naming the dog for the full epic, not just the character concept.
Character Qualities That Map to Dogs
Odysseus is clever, loyal, tenacious, and resourceful — qualities that translate meaningfully to dog personality. The name suits a dog who figures out how to open gates, escapes repeatedly, or otherwise demonstrates unusual problem-solving ability. Border Collies have earned this name through sheer cognitive capability.
The Counter-Reading: Four Syllables Is a Commitment
Odysseus! — you need two full seconds to call this name across a field. Most owners will be using Ody within months, at which point the question is whether the full name matters or the nickname does. The human name Odysseus exists but is extraordinarily rare; the dog version is slightly more plausible.