What did Odysseus give the Cyclops as a gift?
Odysseus gives the Cyclops Polyphemus very strong wine as a gift, but it is not with good intentions. In fact, he gives the Cyclops the wine to put the Cyclops at ease before he eventually passes out from drunkenness. When the Cyclops is passed out, Odysseus retrieves a hardened piece of wood from the fire and uses it to blind the Cyclops as a part of their escape plan from the Cyclops' cave.
This goes against the ancient Greek concept of xenia, or the guest-host relationship. They are supposed to be respectful of each other, giving gifts when appropriate, and that forms lifetime friendships that can often be quite important. But Odysseus defies this very common concept as a way to free himself from someone who had also defied that concept.