David R. George III wrote:
I know that some fans take this exchange to be an indication that Spock has "another name," and most seem to infer that it means that he has another given name. For me, this never quite rang true. Leila and Spock could be speaking about a Vulcan equivalent for his name, perhaps in an ancient tongue, or perhaps they are talking about a family name, or perhaps even some other name he might have had in his infancy or youth, as part of some Vulcan ritual of which Ms. Kalomi is aware but that the audience is not. Also, Spock's response might not indicate that he even has "another name," if he was, as I suggested, answering wryly.
I have to disagree. In the context of the scene, I don't see why she'd ask if he had some esoteric alternate form of his name or some old ritualized thing. The vibe I've always gotten off the scene is that it's a personal moment, she cares for him, and it's awkward to keep addressing him so formally as "Mr. Spock," so she's asking if there's a more familiar form of address she could use.