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So we all know that Tom's nickname name translates to "flight of death" in French.
Which is ironic because he fancied himself immortal, having made his Horcruxes. And it was a nickname he received/made not long after Hogwarts, i.e. not long after he made his first Horcrux and believed himself immortal (He just didn't use it openly until a decade later).
So little Tom was doodling with his name, trying to make something that was in no way reminding of his Muggle heritage, and also remembering the French he glanced at when he learned that it was sort of a "pure-blood" language tradition.
And little Tom, believing himself to be newly immortal, stumbles upon an anagram , a smush of vol-de-mort that means death flies. He sees it as fate, that death flies from him because he is just that powerful as a student.
And he needs to be seen as noble, that he is descended from pure blood, that as a descendant of Slytherin he is the true lord of darkness.
And thus, the name Lord Voldemort was born.