"Important quotes usually depend on the context of what kind of things your are looking for. Most teachers have students read The Odyssey so they can recognize how the main characteristics of our current idea of a hero were outlined in the Greek epic three thousand years ago.
Epithets- these are short phrases that describe a character, place, or object as they name them. Odysseus' heoric traits are often stated in these epithets, and they have helped his name live on. Just a few examples,
"master of of land ways and sea ways" (book 8)
"that man skilled in ways of contending" (book 1)
"the great tatician" (book 7)
Quotes showing god and goddesses interests in Odysseus. This is a requirement to be considered an epic hero.
"Sing in me muse..." (book 1) This is Athena asking for the help of her sister(s) the Muse in order to tell the inspiring story of Odysseus. Athena takes the most interest on the side of Odysseus.
" ‘O hear me, lord, blue girdler of the islands..." (book 9). This is the cyclops Polyphemus praying to his father, the Olympic god Poseidon. This shows two things. One, that another Olympic god is going to play a part in the story, and two, it shows the reader that Odysseus just defeated the son of a great god through brute strength and cunning.
Quote showing the importance of home.
"...you’d stay right here, preside in our house with me
and be immortal. Much as you long to see your wife,
the one you pine for all your days . . . (book 5). This quote is from the minor nature goddess Calyspo. She is obsessed with Odysseus and offers him paradise and immortatlity. Instead, Odysseus wants nothing more than to return home. For the Greeks who often traveled abroad for trade and war, home had to remain the most important.
Quote showing a visit to the underworld and loyalty.
"I lay down on the roof of Circe’s palace, and never thought, when I went down, to go by way of the long ladder, but blundered straight off the edge of the roof" (Book 11). This quote comes from Odyssues man Elpenor in the Land of the Dead. Elpenor's story always makes me laugh, but I don't think it is supposed to. He explains to Odysseus that he passed out on Circe's roof and then feel to his death when he tried to get down. He needs to be buried in order for his soul to be released to the heavenly Elysium fields. This shows Odysseus has traveled through the underworld as is required by epic heroes and that he is loyal to his men, as he promises to stop and bury (burn) Elpenor's body (which he does).