"I know what you are known as," said Dumbledore, smiling pleasantly. "But to me, I'm afraid, you will always be Tom Riddle. It is one of the irritating things about old teachers. I am afraid that they never quite forget their charges' youthful beginnings." (20.157)
What's in a name? Truckloads. Names may be superficial things we attach to ourselves, but they play a huge role in forming our identities. One of the first things we learn about people are their names, and names lead to a whole list of assumptions about where a person is from and what the person is like. Voldemort's rejection of his father's name reveals his desire to cut himself off completely from his family, and reveals his desire to be super scary.
We don't know about you, but we find the name "Voldemort" to be pretty chill-inducing. It has something to do with that "vol" sound – it makes your mouth get all small, round, and kind of like you are about to say something sinister. "Vol" means flight in French. Plus v's are pretty rare – not many names begin with the letter "v" (we can think of Virginia, Vernon, Velma, Veronica, Violet, Voltron, Victor). The "de" in between ensures that this name has three syllables, because one must go big or go home. "De" means "of" or "from" in French. And then, of course, there's the "mort" which means "death" in French. Put them together, and Voldemort has assembled a name for himself that reveals his greatest goal: "flight from death." Obvious much, Voldy?