What do we mean when we say it today?
Chances are when you've heard this phrase before, it wasn't being said by some teenager trying to explain away her secret love for her family's enemy. These days, we use the phrase "what's in a name?" to ask all kinds of super intellectual, high-falutin' questions about life and language. Sometimes people ask this to get to the larger ideas we've been talking about: that language is really just random at some point.
But there are other, more casual times this is used as well. We usually come across this phrase when people are saying that names don't really matter, that all you need to know is what something is, not what it's named. This might apply to designer brands or fancy purses. What's in a name anyway? You can buy the knock off for way cheaper and still have a dress or a purse or whatever. Why should we assign so much meaning to a name (or in this case, a label) in the first place?