I broke up with a girl because of her name
What was wrong with her name?
Her name was so lame. How can someone with such a lame name possibly be good at anything else?
Normally I'd say it is ok to break up with someone if the relationship is not working. However, breaking up with someone simply because of her name does not seem like a valid reason. It has nothing to do with her character or traits.
I don't care, I'm busy and I'm done talking to you.
Well, if you're still here, I have to reiterate that you should not break up with this girl just because you don't like her name.  Her parents gave her that name, and even if it is unintentionally funny or lame today, you shouldn't judge her for it.