1. 네 친구 이름(이) 뭐야? (내 친구 이름은) X야 - casually, between close friends.
2. 친구분 성함이/이름이 뭐예요? (제 친구 이름은) X예요. - formally, between unfamiliar adults.
(성함 is a formal version of 이름, name)
In (2), there is no "your" before 친구분, but it's immediately understood as "your friend" because 분 raises it and this kind of formality is only used for the other person. 당신(의) for "your" is never used in face-to-face conversation, even though some books might use it.