Odysseus's dealings with Polyphemus, the Cyclops, go a long way in terms of showing how Odysseus is both clever and heroic. When it becomes clear that Polyphemus is savage and brutal and will continue to kill Odysseus's men, Odysseus says that he waited until the Cyclops fell asleep that night, and
"then formed the plan within my daring heart of closing on him, drawing my sharp sword from my thigh, and stabbing him in the breast where the midriff holds the liver [...]. Yet second thoughts restrained me, for then we too had met with utter ruin; for we could never with our hands have pushed from the tall door the enormous stone which he had set against it."
First, these lines show Odysseus's great courage, a requirement of all ancient Greek heroes. He was willing to endanger himself by attacking a creature so many times larger than himself so that he could save his men. Second, these lines also prove that Odysseus exercises forethought and intelligence when making plans, even when he is in grave danger. Rather than behave rashly and thereby sound the death knell for his whole crew, he realizes that they need the Cyclops to roll away the stone door, and so he cannot kill him.
Odysseus's new plan to blind Polyphemus is much sounder because being blind will not stop the Cyclops from rolling away the stone door. Odysseus cleverly tells Polyphemus that his name is "No one" or "No man" (depending on your translation), and then gets him very drunk. When the monster passes out, the crew blinds him with a sharpened olive stake, and when the other Cyclopes come to answer his screams, he shouts that "No one" is hurting him and so they go away! Therefore, when Odysseus speaks to Polyphemus and tells him this lie, he is likewise displaying both his cleverness and heroism. He says,
"'Cyclops, you asked my noble name, and I will tell it [...]. My name is No man. No man I am called by mother, father, and by all my comrades.'"
His cunning and clever planning and sheer bravery are reasons why Odysseus is counted among the most heroic figures in Greek mythology.