What names do they exchange? Why is this significant?
I assume you mean what names do Equality 7-2521 and Liberty 5-3000 take at the end of the novel? Equality gives himself the name Prometheus. Prometheus is the Greek Titan in mythology who brought fire that he had stolen from Mount Olympus to human beings. By bringing humans fire, he is also lighting the darkness physically and expanding knowledge mentally. This is significant because not only does Equality discover electricity in the subway tunnels, he is also a symbol of a new man who will lead civilization out of the dark and into a democratic world where everyone will have free choice and individuality.
When Equality meets Liberty, he originally names her the Golden One because of the profound effect she has on him. He sees her sowing seeds in a field and describes her as having a halo or glow around her. Liberty escapes to the forest and meets up with Equality after his flight from the city. They make their way to an abandoned house full of books, clothes, and mirrors. These are something they have never seen before. Gaea is especially taken by the mirrors since she has never looked at her reflection. Equality then changes Liberty’s name to Gaea, the Greek God associated with Mother Earth. According to the legend, she gave birth to the first humans. Not surprisingly, Gaea becomes pregnant shortly after arriving at the abandoned house. She will be the “mother” of this new civilization she and Prometheus will establish away from the oppressive society dominating the earth.
Probably the most significant thing about the two characters changing their names is that they are able to do it for themselves. They were given their names in the society they escaped, and by choosing their own names, they are able to assert their independence and individuality.