Grammar

Franky12What is what is so important for you?

No.

Franky12What about this:What is what that is so important for you?

No. Neither is a possible construction, and I have no idea what you are trying to achieve.

Is that so important for you that you would ignore common decency?

Mister MicawberI have no idea what you are trying to achieve.

I just have those sentences in mind and was unsure about them to be correct or not.

I meant something like: What is the thing that is so important for you? And wanted to use "what" instead of "the thing" here.

What as a interrogative pronoun:

What is that that is so important for you?
What do you call that thing that you brought?

What as introducing a nominal relative clause:
http://grammar.about.com/od/fh/g/Free-Relative-Clause.htm

W

Franky12What is what is so important for you? ... What is what that is so important for you?

Those don't work, but you can make a cleft sentence (which it seems you want):

This is so important for you. (non-cleft version)
It is THIS that is so important for you. (cleft version)

and make the yes-no interrogativ

CalifJimIs it this that is so important for you? (cleft)

Are both "it" and "this" antecedents of "that" in the above cleft question?

Anonymous CalifJimIs it this that is so important for you? (cleft)
Are both "it" and "this" antecedents of "that" in the above cleft question?

No. It is ... that (declarative) or Is it ... that (interrogative) is the framework of all cleft sentences, and the 'it' is non-referring 'it', i.e., "dummy it".
Consequently, only 'this' i

CalifJimonly 'this' is an antecedent of 'that'

Thank you for the reply.

CalifJimWhat is it that is so important for you? (cleft)

I choose this. Thank you.

And the reason for "What is what that is so important for you?" maybe is, it's not standard English, or it doesn't have a right word order or etc, right?

"What is what that is so important for you?"

The word "what" cannot be used in that position.

What is it that is so important for you? (Standard)