Did the interviewer just ask for her name?

It's a pun. "Yuki" also means "snow".

Also a reference to the movie, where Kyon and Yuki invoke the pun at the hospital.
That's gotta be Haruhi holding the microphone.

Moonspeaker said:
It's a pun. "Yuki" also means "snow".

I know, I asked before the translation was up. Guess I guessed correctly.