Odysseus's intelligence is perhaps his most important character trait, as it is what distinguishes him from the other legendary heroes of Ancient Greece. Where others use their brawn—their physical prowess, their political might, their wealth, their speed, their beauty—Odysseus uses his brains and very often gets the better of his opponents as a result. He is the special favorite of the goddess Athena, goddess of wisdom, warfare, and the practical arts, because he embodies the virtues and skills over which she presides.
The Odyssey is full of references to Odysseus's intelligence. Page numbers will vary between different translations and editions of the poem, so to be clear, I have taken all of these quotes from the Robert Fitzgerald translation, and I have included book and line references, rather than page numbers. If you look up the book and line references in your own copy of The Odyssey, you can find the appropriate page numbers for your translation. I will provide some general quotes that describe Odysseus's intelligence and then refer you to the most famous passages in the poem, in which his intelligence is the deciding factor in events. Look up the passages described to get the necessary quotes for your essay—they are all worth reading.
In the first book, Zeus says to Athena:
Could I forget that kingly man, Odysseus?
There is no mortal half so wise.
—Book I, lines 84-85
Later, Athena says to Telemakhos:
But never in this world is Odysseus dead—
only detained somewhere on the wide sea . . .
He will not, now, be long away from Ithaka,
his father’s dear land; though he be in chains
he’ll scheme a way to come; he can do anything.
—Book I, lines 233-242
Odysseus's old comrade in arms Nestor also praises his intelligence:
And as to stratagems, no man would claim
Odysseus’ gift for those. He had no rivals,
your father, at the tricks of war.
—Book III, lines 127-129
When Odysseus leaves Calypso's island, he builds a raft to sail away. Poseidon tries to drown him before he can reach another shore, but Odysseus, despite being exhausted and half-drowned already, thinks quickly and manages to avoid death:
A heavy surge
was taking him, in fact, straight on the rocks.
He had been flayed there, and his bones broken,
had not grey-eyed Athena instructed him:
he gripped a rock-ledge with both hands in passing
and held on, groaning, as the surge went by,
to keep clear of its breaking.
—Book V, lines 441-447
Having landed on the island of the Phaiakians, Odysseus is bruised, battered, starving, and exhausted. He needs help, so when he sees the princess Nausikaa, he approaches her to beg for assistance. He has to gamble on the best way to win her help, and again, quick thinking comes into play:
debating inwardly what he should do:
embrace this beauty’s knees in supplication?
or stand apart, and, using honeyed speech,
inquire the way to town, and beg some clothing?
In his swift reckoning, he thought it best
to trust in words to please her—and keep away;
he might anger the girl, touching her knees.
—Book VI, lines 150-157
Odysseus is often described as speaking "carefully" and "tactfully." His words and his wits have helped him out of many situations, as he tells the Phaiakian court when they ask about his travels. Read through the whole of Book IX for the tale of how Odysseus escaped the cave of the Cyclops by getting him drunk and stabbing his eye out. See Book X, beginning around line 336, for the story of how Odysseus outwits the sorceress, Circe, who has turned his men into pigs and tries to enchant Odysseus as well. Odysseus not only avoids Circe's enchantment, he also makes a powerful ally of her and takes all of her advice to heart when he leaves her island.
Book XII, lines 170-241, describes Odysseus's journey past the Sirens; their song lures sailors to destruction. Odysseus is curious to hear the song and ingeniously manages to avoid the peril of listening to it by having his sailors lash him to the ship's mast. He plugs their own ears with wax so that he can listen but not steer the ship, and they can steer the ship but not listen.
As soon as the crew safely passes the Sirens, they have to sail between the twin dangers of the monster Scylla and the whirlpool Charybdis. Odysseus skirts both dangers by keeping a clear head in an incredibly tense situation, although he still loses some men to Scylla's attack (Book XII, lines 242-312).
When Odysseus finally reaches Ithaca, he must disguise his true nature from the suitors who have taken over his household in order to plan how to get rid of them most effectively. Books XVI–XXI are full of his deceptions and schemes as he determines which members of his household he can count on, what resources he can use, and what the best way to round up all the suitors is, so he can eliminate them all together. I would look at Book XXI, in which Odysseus tricks the suitors into giving him a weapon, which he then uses to kill them.