I am a native Polish speaker. In my native language it wouldn't make much sense to answer the phone by saying either "this is she' or "this is her". Also, I don't believe this phrase was meant to be a short version of "This is she speaking" (or "This is her speaking") as press.uchicago.edu is suggesting in the original post since the correct expression conveying the implied meaning is simply "She is speaking" (and not "Her is speaking" btw). I think that "She is speaking" or simply saying "This is 'insert your name' " is the most grammatically correct way to reply to the caller when answering your phone. However, the expression in question is a very common way to answer the phone in modern English. I use it myself and believe the "This is she" version to be correct. I'll attempt to explain why below. What is present in my native language, was originally present in Old English and has been lost over time in the modern English is the presence of grammatical case. Modern English seems to utilize only 3 forms of grammatical case while Old English and many other modern languages use more than 3 - http://en.wikipedia.org/wiki/Grammatical_case.
When a person answers the phone, whether they (not them) are trying to say that they are speaking or they (again - not them) are trying to equate themselves to the person the caller is asking for they are communicating in the nominative (subjective) case and hence "This is she" is the accurate version. If we were to debate phrases in other than nominative case the answer would probably require some more analysis since for example the modern English case of objective ("her") could correspond to either the accusative, dative or ablative case however there is no doubt that we are debating a nominative/subjective case.