One of the great qualities of Odysseus in the works of Homer is his cunning. Therefore, there are many places in the the Odyssey, which speaks of his cleverness. So, I will have to be selective.
First, the most obvious case of his cunning is his dealings with Polyphemus, the cyclops. When he is trapped with his men in the cave, he is able to escape through cunning. First, Odysseus gets Polyphemus drunk and then blinds him. He also makes him open the mouth of the cave to escape. In other words, he uses Polyphemus' strength against himself to his advantage.
Second, he is also able to overcome Circe with the use of magic. So, when Circe turned his men into pigs, he was able to take her by surprise and make her restore his men.
Third, he is also a master at disguises and strategy. So, when he finally gets back to his home in Ithaca, he is able not only to lay low, but also create the right time for attack. He does this and is able to kill all the suitors.
As the above post noted, Odysseus is known for his intelligence. One of the most well-known demonstrations of this intelligence occurs in chapter 12 of the Samuel Butler translation of the Odyssey when Odysseus and his men encounter the Sirens on their journey back to Ithaca.
The Sirens are mythical creatures that live on an island and attract men to their death with their beautiful and irresistible singing voices. At this point in the story, Circe has told Odysseus what he must do to get back to Ithaca. The route they must take passes by the Sirens’ island. Odysseus knows that they will not be able to overcome the lure of the Sirens’ voices, so he uses his intelligence to devise a plan.
Once the men near the island, he has the men plug their ears with beeswax to drown out the sound of the Sirens’ voices. However, he wishes to hear the Sirens’ himself, but knows that he will not have the power to resist them, so he has the men lash him to the mast. It is a good thing that Odysseus is smart enough to realize his own limitations, because, as we see by the excerpt below, he would not have been able to stop himself from approaching the Sirens, which would have resulted in his death:
They sang these words most musically, and as I longed to hear them further I made by frowning to my men that they should set me free; but they quickened their stroke and Eurylochus and Perimedes bound me with still stronger bonds till we had got out of hearing of the Sirens’ voices.
Odysseus’ brainpower is also demonstrated in the Iliad, which is the prequel to the Odyssey. The Greeks and Trojans have been fighting to a stalemate for ten years during the Trojan War when Odysseus devises the famous ruse using the Trojan horse to conceal his men and trick the Trojans into bringing them inside the impregnable walls of Troy. Today, we still often hear the term “Trojan horse” to signify a trick involving some hidden danger, such as a computer virus.