What significance does Zeus have in The Odyssey by Homer?
This is a good question. Unlike modern books, the Greeks viewed the gods as active agents in society. So, when we come to Homer's Odyssey, we see that Zeus has an important role.
For example, Zeus is the king of the gods and he is the one who reassures Athene, a goddess, that Odysseus will eventually make it home, and that he would make it home with great riches. Here is a quote:
"They will honour him from the heart, as if he were divine, and send him on board ship to his native land, gifting him piles of gold, and bronze and garments, more than he could ever have won from Troy if he had reached home safely with his fair share of the spoils."
Another example of the importance of Zeus is that he releases Odysseus from the island of Calypso. Calypso holds Odysseus as a prisoner. Zeus sends Hermes to tell Calypso to release him. This starts his long journey home.
As the head of the Olympian gods, Zeus usually plays a key role in all the stories involving multiple gods (and often times in his own stories because he liked to father children). He claims that he holds Odysseus in high esteem (Bk. 1, line 63), but his brother Poseidon does not because Odysseus blinded Poseidon's son, Polyphemus.
One big role of his in the Odyssey is his support for his daughter, Athena, and her desire to help Odysseus. He allows her to go to Ithaca to tell Telemachus that his father, Odysseus, will return home (Bk. 1, line 81). He also sent two eagles as a sign to help Telemachus (Bk. 2, line 146). As is stated above, he also makes Calypso release Odysseus and his men (Bk. 5, line 29).
In this book, Zeus' role is a little lesser as compared to his role in other things, but he is still important because he is what keeps things moving along.
In book 9, Odysseus's boat is swept about the sea for nine days by the command of Zeus before it finally comes to rest in the land of the Lotus eaters. On this land, Odysseus men eat the fruit of the lotus plant, turning them into mindless lotus eaters. They lose their will to continue on the journey and wish only to stay on the island,eating the fruit for all of eternity. To try to break the trance, Odysseus and his men force them onto the ship and leave the island with all due haste.
Much later in book 12, Odysseus and his men slaughter and eat the cattle of the sun. The Sun is enraged and prays to Zeus to punish them. Zeus then makes another storm which wrecks the boat and causes the death of all those on board excepting Odysseus, who washes up on the island of Calypso.