Why does Odysseus call himself Nohbody?

The answer 'Nobody' saves Odysseus and his men (who must extricate themselves from Polyphemus's cave, and cannot do this on their own, the boulder stopping the cave's mouth being too large) when the neighbouring Cyclopes come to check out the commotion (Odysseus has put out Polyphemus's eye, and the monster is howling in agony). When the callers ask (in effect) 'Who caused you such distress?', Polyphemus answers 'Nohbody.' So the callers chalk it up to some impersonal cause, and depart. Had they removed the boulder, the jig would have been up.

So far, so good.

The choice of the name is ingenious. It is what we have come to expect from Odysseus. This can't however be the whole story. Ingenuity is one thing. Prophetic power is another. Odysseus, cunning as he is, could not have known what question the Cyclopes would ask. Had they asked 'What's causing you pain?' rather than 'Who did this to you?', the name would probably not have saved Odysseus.

There are two points here, both subtle. The first is that the ingenuity belongs not to Odysseus but to the poet; to Homer, let us say. And this epic is a demonstration of the art of (ingenious) story-telling. (In fact, The Odyssey is much less interesting thematically, much less deep, than the Iliad. It's a good thing that it has the narrative texture that it does.)

A second, and more important point, is that Odysseus, after the (Trojan) War is over, does not have a clear identity. The end of the War is the end of the warrior way. What are men and women after that way, the dominant way of the whole culture, is obsolete? In the interim, they lack identities. They are nobodies. So, Odysseus's choice of name also captures an essential feature of the epic's essence, that it explores basic issues of identity (in that cultural context). The exploration, in my judgement, is pretty monochromatic. But others may think differently.