What is an example of something Odysseus should have brought on his adventure in the Odyssey?

In one sense, given all his adventures and the dangers of the sea journey, he might not have wanted to take anything valuable. However, if he thought he could avoid having it stolen or destroyed, his great bow, which he used to slaughter the suitors at the end of the poem, would have been useful.

Odysseus was traveling through wild lands in which he and his sailors needed to hunt in order to feed themselves. A powerful bow is a very useful hunting weapon. It also works well for self-defense. Even though the bow was sometimes considered a coward's weapon in the Trojan war (Paris, who is primarily an archer, is considered a coward), it still would come in useful against monsters and wild animals.

Some sort of fire-starter would also have been useful as he and his men were spending a great deal of time camping.