It's "This is she." Here's why:

The 'is' is like an equal sign in this instance, because sentences like this, "is" shows the state-of-being relationship between the two ideas. "She" is identifying herself in this sentence. 'This' and 'she' are the same thing, and therefore are in the same case (the nominative). It has the exact same structure as "I am an English teacher." "I" and "an English teacher" are both nouns, and both in the nominative case, and are both the subject of the sentence, although in English you can't really reverse them and say "An English teacher am I." Unless you're Yoda.

Someone earlier talked about "This is her" as being possessive. That's correct, in a sense. To say "This is her" you must add an object to the sentence, as in "This is her ball."

You cannot correctly say "This is her speaking," because speaking is not a noun, and therefore cannot be an object. You *could* say "This is her ball," because "ball" is a noun and can be an object. But she OWNS the ball, which is an object.