David R. George III wrote:
^ Well, I'd say that wasn't really clear either. That exchange in "Journey to Babel" went as follows.
Kirk: Mrs. Sarek, I just don't understand.
Amanda: Amanda. I'm afraid you couldn't pronounce the Vulcan name.
Kirk: Can you?
Amanda: After a fashion, and after many years of practice.
This might suggest a family name to some, but I never thought so. For one thing, Amanda tells Kirk that he "couldn't pronounce the Vulcan name" after asking him to call her by her given name. I thus assumed that the "Vulcan name" of which she spoke was an analogue of her Terran given name.
Again, that doesn't make sense to me in context. Kirk has just called her "Mrs. Sarek," i.e. assuming that "Sarek" is her married surname. It therefore logically follows that the "Vulcan name" she's referring to is her true married surname, i.e. Sarek and Spock's family name.
One can imagine any number of in-universe explanations for Leila Kalomi's question of Spock. Back when they knew each other on Earth, where she had fallen in love with him, perhaps their conversations had turned to Spock's life, and he had intimated that he had "another name" without ever revealing it. Or perhaps he'd told Leila that some Vulcans took other names under specific circumstances or for specific reasons. There are uncounted ways such conversations could have set up her years later saying, "You never told me if you had another name, Mr. Spock."
That's overthinking it. In the context of that moment in the story, there is no reason why she would engage in such a total non sequitur
as to bring up something from years in their past. The emotional tone of that moment in the episode is that she's frustrated at only being able to call him by his formal designation, "Mr. Spock," and is probing to see if he has a more personal, familiar form of address.
I profoundly doubt that D.C. Fontana, a working Hollywood scriptwriter who had to churn this episode and "Journey to Babel" out in a matter of weeks while busy with all sorts of other stuff, had the time or the inclination to concoct all sorts of convoluted and arcane worldbuilding that she was secretly alluding to when she wrote these lines. She was writing these scripts for a 20th-century American television audience, and she would naturally have written them to reflect the cultural references and expectations of the culture she belonged to and wrote for. If she wrote a scene where a woman refers to a man she loves as "Mr. X" and then asks if he has another name, she most likely meant a given name, a personal name as opposed to a surname. If she wrote a scene where a man referred to a woman married to foreign dignitary Y as "Mrs. Y" and the woman offered him her first name as an alternative, explaining "You couldn't pronounce the [foreign] name," she was most likely thinking of Mr. Y's true surname.