Throwing in my two cents a bit late here.
For starters, I usually call him "Dr. Robotnik" or "Dr. Eggman" interchangeably, unless I'm talking about the old Western content like Archie or the cartoons, which then I'll just call him Robotnik.
But I actually like the name Eggman with the context it's given. That, it's a nickname Sonic made up to mock Robotnik, but then the doctor decided to instead own that name as a sort of power play. His most famous theme song has the chorus "I am the Eggman!", for crying outloud, he took that name in stride.
Plus, I feel as if people completely miss the menace factor Eggman still has, it's just that he's much more sinister and not so blatant with it like in the old Western cartoons.
Sure, he's constantly got that big funny grin on his face, but that's what makes him terrifying. He's a sadistic mad doctor who'll gladly indulge in the schadenfreude of your demise while he plasters that same grin everywhere he can, and suddenly, that it's not that funny of a grin anymore.
The best way I can describe Eggman's creep factor is that Eggman is scary like how clowns are scary, if that makes sense. Think Eggmanland, a death trap of an amusement park filled with bright colors, and most importantly, his imagery all over the place. Seems goofy at first until you realize Eggmanland is the pinnacle of Eggman's cruelty. Designed to have your last moments be that of wacky contraptions plastered with his mug as if to laugh in your face as you draw your last breath.
That cruelty is a big factor for Eggman, too. I've already mentioned death traps made to infiltrate your mind with his image, but he'll also destroy nature, imprison wildlife, force the world around him to conform to
his vision. Let's not forget that time he held Tails and Amy hostage as a way to get Sonic to hand over a Chaos Emerald he knew was fake, having a gun pointed at Amy fully intending to paint the floor red and pink, and subsequently launching Sonic in a shuttle into his supposed death right in front of his two dearest friends.
And on top of all that, he just
towers over everyone else in the cast. He can, will and
has just tossed these oversized animals around like ragdolls from his sheer size alone, and you gotta speculate he does have some strength to him.
I may be going too in-depth on a thread just asking if you prefer the name "Robotnik" or "Eggman", but talk here seems to imply that the Eggman we have today is just a joke compared to old Western Robotnik, and although modern Eggman has his comedic moments, him being mainly played for laughs is FAR from the case.
He only calls himself that for the satisfaction of seeing you bow down before the one you used to jokingly call "Eggman".
This is the man who blew up half the moon as a threat.
He is Doctor Ivo Robotnik.
But Most importantly...
HE IS THE EGGMAN!!!
(And sometimes Robotnik.)