Word order in an embedded question ("She can't decide which IS the best place for it")
Solution 1:
Normally an embedded WH-question uses statement, not question, grammar, like your second example:
Tell me what his name is.
or
I don't know when it starts. (not "when does it start")
and indeed your first example would be grammatical that way:
She can't decide which the best place for it is.
But the form you have given is more comfortable. I think this is because of "heavy element extraposition": since the complement is four words long, and the verb just the little word is, they are swapped back, to leave the verb next to its subject.
Edit: I had my description wrong, referring to "subject-verb inversion", which is not what is at issue here. Thank to Sumelic for pointing this out (after two and a half years!)
Solution 2:
There are two different factors that can cause "unusual" word order in clauses with a wh-word:
Wh-fronting: usually, the wh-word comes at the start of the clause. (Sometimes other words in the phrase "move" with it: this is called "pied-piping")
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Subject-auxiliary inversion: in non-embedded questions where the subject is not a wh-word, we put an auxiliary before the subject (e.g. Who am I?; "am I" is AuxS word order). "Do-support" is used to provide an auxiliary if there isn't already one in the sentence (e.g. What did they do?; "did they" is AuxS word order).
Subject-auxiliary inversion does not occur when the subject is a wh-word (e.g. "What was in the box?": the subject what comes before the auxiliary was).
Subject-auxiliary inversion is typically not used in embedded questions (e.g. I don't know who I am, Do you know what they did?).
Your examples
In "Tell me what his name is", we have SAux order with "[his name] [is]"; the predicative complement "what" comes before this because it is a wh-word.
In "She can't decide which is the best place for it," I can think of two possible explanations for the word order in "which is the best place for it". The wh-word "which" might be understood to be the subject, in which case, we would have SAux order here. Another possibility is that "the best place for it" is understood as the subject, and we have AuxS word order. Subject-auxiliary inversion in embedded questions is not standard, but it is a feature of some dialects of English according to the Yale Grammatical Diversity Project: English in North America.