- 이 사람(의) 이름은 뭐예요?
- 이 사람(의) 이름이 뭐예요?
(The possessive 의 is usually left out without affecting the meaning)
There is a subtle difference between the particles 은/는 and 이/가, although you can use these two sentences interchangeably in most cases.
In simple terms, both 은/는 and 이/가 can mark a noun as the subject of a sentence, but the topic marker 은/는puts a greater focus on it, so you use it when you want to introduce, explain, or describe what you think in your own way.
The subject marker 이/가 is for when you respond to some external situation rather than starting from your own prepared thoughts.
For example, compare these.
- 나는 남을 도와주고 싶다 (I'd like to help others) => explains something about yourself(나), the topic, so 은/는.
- 제가 도와드릴까요? (Shall I help you?) => The focus is on helping the person (the situation), not the subject 제가, so there's no need to use the topic marker.
(In these examples, switching 은/는 and 이/가 will make the sentences unnatural)
Going back to your sentences,
- 이 사람 이름은 뭐예요? => implies curiosity about the person's name (and perhaps the person too).
- 이 사람 이름이 뭐예요? => sounds more business-like, as if there's some practical need to know it.
(In this case, the difference is a rather small nuance thing.)