How to say "This person asked me, 'what's your name?'" in Korean

Question

Answer

Translated it's "누가 제 이름이 뭐냐고 물어봤어요." 
Literally meaning, "Someone asked me what my name was."
If you want a closer translation to your sentence, then it's "누가 저한테 물어봤어요, '이름이 뭐예요?'"
But the first one sounds more natural if you're talking to someone.
Hope that helps!

Answer

이사람이 '너 이름이 뭐냐?'고 물어봤어요.

Answer

그 사람이 저에게 이름이 뭐냐고 물어봤어요

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