Why are Death-Eaters called Death-Eaters? What does their name mean?
Voldemort's followers were called "Death Eaters" because one of Voldemort's key goals was to use magic to fully control Death, directing its powers at Voldemort's sole command. Voldemort sees this as the sign of ultimate power, which is his foremost goal; to be the most powerful sorcerer ever.
Obviously, killing (directing Death toward others) is easy; it's preventing Death from turning on you that's tricky. The third leg of the triangle formed by the Hallows, recalling others from Death, was never of interest to Voldemort, and even the Resurtrection Stone didn't really do that.
The name, therefore, arose from what being one of Voldemort's disciples means at a fundamental level. It means that you believe in Voldemort's philosophy, that mortality is a weakness of lesser beings, and that to become truly powerful, you must fear neither dealing death or succumbing to it; in the vernacular, you "eat Death for breakfast".
According to J.K. Rowling, the Death Eaters rose out of a group known as the Knights of Walpurgis, and she shared this in an interview with the BBC:
"'...in here is the history of the Death Eaters and I don't know that I'll ever actually need it - but at some point - which were once called something different - they were called the Knights of Walpurgis...'" - J.K. Rowling.
BBC Newsnight, Thursday, 19 June, 2003, J.K. Rowling: The Interview via Mugglenet
The Knights of Walpurgis was a play on Walpurgis Knight: Walpurgis Night (in German folklore) the night of April 30 (May Day's eve), when witches meet on the Brocken mountain and hold revels with the Devil... (Oxford Phrase & Fable) The Knights of Walpurgis by Cindy Eric at Mugglenet.
Anyhow, that's the history of the Knights of Walpurgis, who preceded the Death Eaters. The most common theory I've heard regarding why Voldemort's followers are called the Death Eaters is that it is a reflection of Voldemort's self-loathing of his own status as a half-blood. Like Hitler, Voldemort set out to obliterate all those who exhibited traits he hated in himself. Why his followers are called Death Eaters specifically is, as far as I know, not addressed by J.K. Rowling anywhere. And how the name "Death Eater" arose from "Knight of Walpurgis" is equally puzzling.
Their name comes indirectly from Christian symbolism. (See the books of John Granger for a thorough exploration of this symbolism in the Harry Potter books.)
There's a long Christian tradition of calling the Body and Blood of Christ, the Eucharist, "the medicine of immortality"; it goes back to St Ignatius in the last first or early second century. If the Eucharist is the medicine of immortality, then those who eat it are partaking of life, and could be called Life Eaters. Then those who pursue immortality by another course, by way of murder and evil, could be contrasted by calling them Death Eaters.
They were called ‘Death Eaters’ because Voldemort’s life goal was to kill muggleborn witches and wizards before he changed his interest to Harry Potter. His followers killed anyone in their way or anyone who stood up for HP. (Not everyone but you get the point.) I assume that their name ‘Death Eaters’ was because they killed lots of people which was effectively like eating because they did it frequently? And they ‘ate’ death by doing it a lot. Sorry if this makes no sense. If anyone knows what I mean and can edit it to make more sense that would be greatly appreciated.