There are (at least) 3 things that you’re born with that you are part of who you are now, that you had no control over:
Your name, your age, and your nationality. These were decided for you. You were thrown into them, and as Sartre would say, they’re your facticities.
Which makes them meaningless in any truly significant way, as to who you really are.
So when Juliet tells Romeo she doesn’t give two fucks about what his name is, she is rebelling against the absurdity of those uncontrollables.
Here’s what she says to him, in Act 2, Scene 2:
‘Tis but thy name that is my enemy.Thou art thyself, though not a Montague.What’s Montague? It is nor hand, nor foot,Nor arm, nor face, nor any other partBelonging to a man. O, be some other name!What’s in a name? That which we call a roseBy any other word would smell as sweet.So Romeo would, were he not Romeo called,Retain that dear perfection which he owesWithout that title. Romeo, doff thy name,And for that name, which is no part of theeTake all myself.