There are different ways to analyse the grammar, but one common way employed is to say that this sentence has a Topic - Comment structure.
In a topic - comment sentence, the topic does not necessarily need to be an argument of the verb (it can, but doesn't need to). What it does is to establish the context of the sentence. Then the comment is a syntactically correct sentence with a predicate, what makes sense in the context that the topic provides.
In Korean, the topic of the sentence is marked with the ending -는. Now, in Korean it is possible for that topic to also be an argument of the verb - to be a subject or object, for instance - and in fact, in most cases, the topic is also the subject - but it does not have to be. It can be, in a sense, syntactically independent. As some answers explain it, in school grammar the topic is considered the subject of the comment, which in its whole is called a predicate. I think this is perhaps a way of analysing it so that the topic is not independent; either analysis works well.
So in the sentence 당신은 이름이 뭐예요?, the comment is just 이름이 뭐예요?, meaning "what is the name?" This doesn't make sense on it's own, without any context - but the topic of the sentence, 당신은, provides the context. We are talking about "you" (당신), and so the question 이름이 뭬예요? provides context - not merely "what is the name", but "for you, what is the name?" - that is, "what is your name?"
We can see other examples of where the topic - comment structure is used with an independent topic:
철수는 머리가 좋지 않다 (Literally: Cheolsu, head isn't good; which means "Charles isn't too smart"). The context (Cheolsu) makes the comment (head isn't good) make sense.
나는 비행기 타는 것이 무섭다 (Literally: I, riding a plane is scary; which means "I'm afraid of flying in a plane".
영희는 계산이 빠르다 (Literally "Yeonghee, calculation is quick"; which means "Yeonghee is good at numbers/calculation".
S 사업은 문제가 많다 (Literally "S Company, problems are many"; which means "S Company has many problems".