This is a really dumb argument. Actually, it goes far beyond dumb. It's fucking stupid and racist as hell.
Leaving aside the entire (well researched and empirically-supported) concept of how microaggressions operate and why they're important to understand*, it's still easy to grasp why this goes beyond some mean words. So easy, in fact, that to make the argument these kids "just need thicker skin" and to "ignore their bullies" is an argument so obviously bullshit that I can't even accept it's simply a clueless ignorance of what's going on, but is instead an intentional attempt to minimize an incredibly important issue.
Because the "mean word" in this case is not just any random word. It is "nigger."
This is a word I typically avoid using even in an academic context such as this, but in this case, it's really important to literally spell out the invective being hurled here.
Because "nigger" is not a mean word like any other word. "Nigger" has a very special place in American history, as the word (and much more importantly, what the word represents) has been integral to maintaining America's racial hierarchy for centuries.
"Nigger" is the reason white men were free to rape Black women for centuries. "Nigger" is the reason it was not only acceptable, but obviously natural and even moral, for Black people to be enslaved. "Nigger" is the reason Black men were lynched simply for the sake of being Black men. "Nigger" is the reason a 14 year old kid could be murdered for supposedly whistling at a white woman.
And it's not like this stuff is ancient history. "Nigger" is the reason a 12 year old kid playing in the park could be murdered by a police officer and the murder deemed "justifiable" by the courts. "Nigger" is the reason a 17 year old kid could be stalked and murdered by a man with a long history of violent crimes and yet somehow be the one who's actions were most closely scrutinized at trial. "Nigger" is the reason the deadliest hate crime against Black Americans in 75 years happened just a few months ago, not the very distant past. "Nigger" is the reason more unarmed Black Americans have been killed by police this year than were lynched in any year since 1923.
So when a white person calls a Black person "nigger," they're not just saying a mean word. They're invoking literal centuries of horrific crimes against Black people. They're invoking the idea that white people can do whatever they like to Black people with the confidence that no punishment will come to them. They are simply employing a short hand way of saying "Not only do I think you're subhuman, but I could murder you right here and right now for no reason and the majority of this nation would spend their time examining what you did to invite this murder, many of whom would celebrate me as a hero." They're invoking the idea that Black lives truly do not matter to a very large segment of our population.
So if you truly can't wrap your head around why these folks are so upset by a "mean word," in your head just replace "mean word" with "being threatened with murder by someone who will face no consequences for their actions," and then see if you still think it's no big deal.
*But for the record, we shouldn't put that aside! The only reason I'm skipping it here is to avoid this devolving into a discussion of whether or not microaggressions are real (they are), because this is one of those terms asshole white people have picked up on as a marker they need to virulently disagree with whatever point is being made, regardless of its merits.