Eponymous hairstyles are hairstyles that are named after a person or a character. Let's say you have a friend named Sally who always wears her hair in two braids. That can be called a "Sally braid." Or, if you have seen the movie "Star Wars" and notice that Princess Leia has a distinctive hairstyle with two buns on the side of her head, you might call that the "Princess Leia bun."
Basically, when a hairstyle becomes really popular because someone famous, like a celebrity or a character in a book or movie, wears it a lot, people sometimes give that hairstyle a name to describe it. That way, if someone wants to ask for that specific hairstyle when they go to a hair salon or ask their friend to do their hair, they know what to call it.