“Is her name Eliza?”
"No. It isn’t OR She isn’t

Which one is correct?

‘It isn’t’ is the best option there as it answers the question directly.

‘She isn’t’ would be better suited to the question, “Is she Eliza?”

If we address 2-person, for example

"Is your surname Anderson?" 

 1) "Yes, I am"      OR

 2) "Yes, it is"

Which one is correct then?

The correct one is (2)