I find it interesting that the tone of argument that 'it's not such a big deal anyway', or 'why are you making such a fuss about nothing', is so often trotted out when the issue being discussed is one relating to women. Women's concerns are merely trivial. Men's (even if they are trivial) are important. Cf. shopping versus football (which should not be 'gendered' according to lazy stereotypical thinking, but are).
'It's just a name' implies similarly - because it's women and not men who are expected by social convention to discard our own identities. This is not mere trivia. Casual, everyday sexism is an issue worth fighting. The expectation that a woman will disappear at the point of marriage at the very least deserves scrutinity. Likewise, whilst I can't understand what compels any woman to take on someone else's name, if she does so then that name becomes hers. I find the expectation that on divorce she will immediately hand back that name, as though it's on loan and once she no longer 'belongs' to the man that name is a possession to be returned, equally unfathomable.
If names didn't matter then I could address you as Dick, Engelbert Humperdink or SeeYouNextTuesday, and you wouldn't bat an eyelid. People would not get so offended if someone inadvertently misspells their name. A family name is an identity, no matter how much those who would divest women of this privilege try to claim the contrary. It's an identifier which lets you pass between countries, marks you out as individual, denotes your origins for better or worse, or which labels your achievements - hence academics who have published and have a particular name on their degree certificates are very reluctant to divorce those things from their own names.
The upshot on this thread, as ever, is that men possess their own names. Women are deemed not to, and far too many women seem willing to support the patriarchy in that antediluvian view. You're of course at liberty to dismiss these things as trivialities. To me, they happen to be very important.