Do we say
"can you tell what your name is?" or" Can you tell me what's your name?'
I would use Can you tell what your name is?- it's the form considered correct as it's an embedded question, but the second is the sort of thing you'll hear people saying in casual speech.
ONLY A NON-TEACHER'S OPINION
(1) I think that learners would be wise to follow the Editor's suggestion.
(2) For advanced learners (and even native speakers), may I report something quite
fascinating that I learned while browsing the great Henry Fowler's masterpiece A
Dictionary of Modern English Usage (1965 edition, edited by Sir Ernest Gowers).
(a) Mr. Fowler opined that "Explain what are the duties" and "Explain what the duties
are" could both be considered "correct," depending on the answer. I shan't bore you
with his reasoning. (If interested, you can check out his book yourself.) I only wanted
to very respectfully and humbly remind learners (and ordinary native speakers like
myself) to be very careful before we rush to ridicule something as "wrong." We may
be the ones who are wrong.