What's your name?
 

Hello,
Is it correct to say "What's your name?"
In Japanese:

そなたのなまえはなにですか。

sonata no namae wa nani desu ka.

That looks like "What is thy name?" to me.

You can just say
お名前は何ですか？
(おなまえはなんですか？) - What is your name?

お名前は？
(おなまえは？) - Your name?

And there are probably others, but these two are common formal and informal.

I just used sonata as a pronoun which means "you" and used no after it that it belongs to you
Just like watashi?

わたしのなまえはなんですか。
watashi no namae wa nan desu ka.

OK can I say nani which means what?
なまえはなにですか。
namae wa nani desu ka.

I don't know why they use nan.

Look at this website, in the question they used "nan".
and in the vocabulary list on the next page they state "nani" as "what".
Vocabulary, Grammar - Lesson 2 - Language - Kids Web Japan - Web Japan

First, sonata sounds really strange. So I just wouldn't use that, at least for what you're trying to say. In Japanese, there are words for "you," because sometimes there's just no way around it, but it's not common to use them, because it's considered too direct. For example, you could say "あなたのなまえはなんですか？" But あなた (you) is direct, so it is commonly omitted and instead it is more natural to simply say "おなまえはなんですか？"

わたしのなまえはなんですか？ doesn't make sense because it's asking "What is my name?"

"Nani" and "nan" kind of conjugate depending on what you're talking about. The kanji for them is the same "何" but in this case the sentence should be おなまえはなんですか？But both "nan" and "nani" can mean "what?"

僕の名前話ジョンです。
Boku no namae wa John desu....

Sonata is an old way of saying 'you'. It's also fairly posh sounding. You'll mostly hear/ read it being said by sophisticated or high social status characters in books/movies etc based in the past.

That's not right.

I've heard that said many times more than "watashi wa .... desu". What's the incorrect portion?

The kanji. Just leave it as "は" - "wa"

So, do you mean that we should always use "nan" before the verb "desu" in the question?