"How should I address you?"
"Monsieur," replied the captain, "I will merely be Captain Nemo for you." (1.10.45-46)
"Nemo" is the perfect name for a mysterious sea captain; it is Latin for "no one." So, by telling Aronnax to call him Nemo, our nutty captain protects his identity… in addition to making himself sound way cool. When you think about it, Nemo really represents all wronged men, then. He is both no one and everyone.
The name "Nemo" also harkens back to a masterwork. No, not the one about the little fish who lost his way, the one about another man who left to wander the seas: Homer's Odyssey. In the Latin translation of this epic poem, Odysseus tricks the Cyclops by calling himself Nemo. So the Nautilus's Captain is not the first world-traveler who throws others off his trail by naming himself after no one.
Oh, and there's one more, even more obscure reference here. See, Verne was really big on Scotland. And the motto of the Scottish royal family happens to be "Nemo me impune lacessit," which translates as, "No one assails me with impunity."
To put it more simply, the Scottish royal family wants to let the world know: "If you mess with us, you better watch your back." It'd be hard to come up with a better slogan for Nemo and his Nautilus, given the captain's ragey obsession with revenge and all.