The stories The Odyssey and the Epic of Gilgamesh share some elements that suggest one influenced the other. Considering that Gilgamesh is older, it would seem that the Odyssey drew inspiration from it. Of course, just because there are parallel parts between two stories doesn't mean they were intentional.
Martin West, an expert on "the classics," has studied the subject. He noted some strong similarities, which I'll highlight, along with others:
- Both characters are strong heroes that have legendary status,
- Both were created as epic poems,
- Both interact with the gods,
- Both stories include aspects of the ancient world's culture, such as guest/host relationships,
- Telemechus helps Odysseus kill the suitors, while Enkidu helps Gilgamesh kill Humbaba and the Bull of Heaven,
- Humbaba, the uncivilized and evil giant, is sort of the Cyclops.
- Odysseus and Gilgamesh travel great distances during the course of their adventures,
- Both, during their journeys, go to the land of the dead,
- To reach the underworld, Odysseus follows instructions from Circe, daughter of the sun-god. Gilgamesh gets instructions from the goddess Siduri, whose home is associated with the sun.
- Circe's island is located at the edge of the world, same as Siduri.
I hope these ideas give you a good place to start in your quest to find other items to compare these stories.
In both the Epic of Gilgamesh and Homer's Odyssey, mortals combat some gods and are aided by other gods. Gilgamesh who, like Odysseus, travels a long way from home, tries with his friend Enkidu to enter the forest forbidden to mortals. To do so, he must fight a monster who belongs to Enlil, god of the earth, and he is aided by the sun god, Shamash. Similarly, Odysseus must do battle with the Cyclops, the son of Poseidon, god of the seas, and he is aided by Athena, the goddess of wisdom, in all his travels. Gilgamesh, like Odysseus, must fend off goddesses. Gilgamesh refuses the attentions of Ishtar, the goddess of love and war, and Odysseus decides not to stay with the semi-immortal nymph Calypso in favor of returning to his wife, Penelope.
Like Gilgamesh, who learns about friendship but loses his friend Enkidu, Odysseus befriends the men who are returning from the Trojan War with him. Odysseus also loses the men on his ship and suffers greatly. After returning to Uruk, Gilgamesh becomes an upstanding leader, dedicated to improving the lives of the people he rules. Similarly, Odysseus, after his travails and his return home, becomes again a respected leader of Ithaca and a devoted father and husband. Both men are mortal but are heroes among men.