You are mistaken. Names were awesome and unique, kids didn't care. Frank's comments during a 1975 interview.

Frank Zappa Talks About His Kids’ Names

In a 1975 interview, Frank Zappa talked a bit about his kids’ unorthodox names:

Cohen: Is one of your children named Moon Unit?

Zappa: Yep.

Cohen: Son or daughter?

Zappa: Daughter.

Cohen: What are your other children’s names?

Zappa: Dweezil and Ahmet.

Cohen: Is Moon Unit the oldest?

Zappa: Yep.

Cohen: What do you call her around the house?

Zappa: I call her Moon and I call Dweezil ‘Dweezil’ and I call Ahmet ‘Ahmet.’

Cohen: I’m wondering what psychological effects a name can have.

Zappa: Well, first of all, I knew that they were going to be unique anyway because of certain other attributes, so why not have a name that goes with it. They all like their names and the kids at school do too. They don’t make fun of them; in fact, most of them are jealous of their names. You know, it always amazes me when someone who is in my age group, or even younger, asks me a question like why did you name your children that. It’s a reactionary kind of question. Why the f*ck not name your kids something like that? They’re having a good time. Besides that, if they ever wanted to change their names they can do it. It only costs about $15.

Cohen: You can’t tell from the name Moon Unit whether it’s a boy or a girl. Had Moon Unit been a boy, would you have named him the same thing?

Zappa: No, I would have named him Motor Head. 

Frank’s fourth child, Diva, was born in 1979.

@Mark013 if your name is really Mark you don't understand. Having to repeat and spell your name every time you say it your whole life is a pain in the ass. Parents don't care, they just like the name, they don't have the problem. If the kids were bullied would that change your mind? This bullshit about $15 to change your name is exactly bullshit because you have to go to court to change your name because you can't avoid what you did with your previous name. I'm sure they also had pretty nice lives with as much money as they had, that makes it a little easier and also a reason why kids wouldn't bully them or make fun of them.

@lerlo My name is Mark. Even my name was made fun of. Mark, Mark, it's the sound a dog with a hair lip makes. My last name was even worse and also made fun of but not going to mention it here but it was one that was always mispronounced and had to be repeated. I never had a problem with the kids making fun of my name and I did it right back, it's what kids did and still do, I drive school bus and hear it now and then. Making fun of a name and bulling are apples and oranges, can't be compared. You can make fun of any name first or last. Just because their names were unique doesn't make it better or worse than a regular name to make fun of. As for the $15 name change, it was 1975 after all and was probably close to the truth at the time before the courts and legal system became profit centers.