In Homer’s world, you live once, badly, and die. The gods are immortal; you are not. The only way a mortal man stretches past his lifespan is kleos — the report of his deeds, repeated by other men, surviving him.
This is why Achilles chooses a short life and lasting fame over a long anonymous one. It’s also why Odysseus, in the underworld, hears Achilles say he’d rather be a slave’s slave than a king of the dead. Once you’re across, kleos tastes like ash.
Odysseus’s relationship to kleos is more complicated than Achilles’s. He’s the hero who survives. Twice he hides his name to stay alive (cyclops cave, beggar in his own hall). He chooses nostos over kleos — and Homer’s quiet argument is that it might be the harder, better choice.