Upon answering the telephone, the person calling asks if Joan is available. If Joan is the person who answered the phone, should she say "This is her" or "This is she"?
Traditional grammarians prefer the nominative ("she") for the complement of the verb "to be". Most usage in my experience prefers the accusative ("her") and regards the verb as having a direct object rather than a complement.
I suspect the traditional grammarians, as they often did, have misapplied a rule of Latin grammar. In Latin, "esse" takes a complement in the nominative case, but Latin declines the verb strongly enough that it doesn't bother with a pronoun as the subject of a verb unless needed for emphasis. "It is she" in Latin would be "illa id est", which looks far more natural than the English.
Note that it's "c'est lui" in French, so there isn't a general rule for a complement of "to be" being in the nominative.
A normal (transitive) verb, like say "have" has a direct object, which is in the accusative case. So, for example, "I have her" uses "her" as a direct object, and "her" is in the accusative case, where "she" is in the nominative case.
In Latin, the verb "esse" ("to be") is special; it doesn't have a direct object in the accusative case, it has a complement in the nominative case. English grammarians in the eighteenth and nineteenth centuries adopted a number of rules of Latin grammar into English; this was one of them. In Latin, "it is she" and "she is it" are both the same thing "id illa est" or "illa id est" can both be translated either way - the point being that the "is" ("est") just equates to things to each other - it's like in maths you can have x=y or y=x and they both mean the same thing.
English, though, takes word-order very seriously, and a pronoun after a verb is very strongly marked as being in the objective case and you don't get the benefit that you get in Latin from the exception - there's still one before the verb and one after; you don't get to make clear that it's commutative. So you have a special-case for a verb, which you get no useful benefit from. It's hardly surprising that most English speakers have reverted to "it is her" rather than "it is she".
A note on cases: Latin is conventionally described as having seven cases (though only five are actually different for most nouns and pronouns). It has two cases, accusative and dative for the objects of verbs. English's residual case system only has the one case for both, which merge into the objective case (in fact, a third Latin case, ablative, merges in too). That's why the two paragraphs above appear to use two different words for the same case.
It's correct either way. English has lost its case system almost completely. This makes it hard even for native speakers to decide between subject case (nominative) and object case (formerly accusative/dative).
A long time ago – far too long ago to be directly relevant today –, English still had a 'proper' case system and the copula be was followed by subject case. This is normal for Indo-European languages. Therefore, originally it would have been something like (translating to modern) "It is she".
However, just like French but under the radar, English developed emphatic forms of its pronouns. When an adult asks a bunch of children "Qui veut de la glace ?" / "Who wants some ice cream?", the children's natural response is "Moi ! Moi ! Moi !" / "Me! Me! Me!", not "Je ! Je ! Je !" / "I! I! I!" This doesn't mean that the response is in dative / object case but that the response is emphatic. The same emphatic form of the personal pronoun is also used in connection with the copula être / be: "C'est moi." / "This is me."
The French third person feminine pronoun happens to have no special emphatic form. It's elle whether emphatic or not. In English, the non-emphatic subject case is she, and all other forms (object case and emphatic form) are her. Therefore, in natural English the correct answer is "This is her." This is how non-native speakers learn to say the sentence.
For some reason, there is a tradition in English grammar (as taught to native speakers) to completely ignore the issue of emphatic pronouns and to assume that English has to work like German, which does not have them, or like Latin, where only emphatic pronouns exist and these carry the case distinctions. A non-native speaker who missed the drills for the emphatic pronouns but who, coming from another Indo-European language, knows about subject case after be, will quite logically say "This is she." English prescriptivism, in its trademark cluelessness (often pointed out by proper linguists), stepped into the same trap. English has a tradition of rules being made up by ignorant prescriptivists and then taught to generations of students and used as shibboleths that serve to distinguish those who received a 'proper' education from those who just use the language naturally and correctly, such as intelligent members of the working class and of course the likes of Charles Dickens and Jane Austen. As a result, "This is she" became a correct alternative again, long after it had fallen out of use.
(Note on case terminology: English still has subject case and object case, though it only marks them on pronouns. It also has genitive pronouns, but since the former genitive suffix 's is now a possession-marking clitic, the status of genitive as a case in English is even less clear. English subject case is exactly the same thing that is called nominative in German, Latin etc. English object case is the result of a merger of accusative and dative, also known as accudative. This merger has long been completed in English, has almost completed in Dutch and many German dialects, but is still in progress in standard German.)