What is the full name for Sue?

"Sue" is a nickname, and there isn't a single full name that it universally corresponds to. It could be short for any number of names, such as:

* Susan

* Suzanne

* Suzette

* Susanna

* Sue Ellen

* Beatrice (sometimes nicknamed "Bea" which can be shortened to "Sue")

It's impossible to know the full name without more context.