Translation:He had often called me his best student.
One could think there could be an issue with translating Schüler as "student" instead of "pupil". So far I remember, DL kind of made a point of reserving "student" as a translation for Student. But I think DL doesn't take this distinction rigidly, as even they themselves used "student" in this translation.
Anyway, I have written "He had frequently called me his best pupil", but it was also not accepted.
The only remaining explanation would then be in the adverb used, but I think there is no significant difference between the meanings of "frequently" and "often" to justify it. In this discussion about this difference, someone tries to reason that "frequently" would imply some sort of regularity which would be absent from the "often", but this doesn't make sense to me. I think to imply regularity one would used "regularly" instead! But I'm neither a native speaker nor an expert. Maybe someone who is can shed some light on this topic?
Perhaps like us, they have two words. Although they are interchangeable, "häufig" is perhaps more directly used as "frequently" and as the adjective "frequent" as well. http://dictionary.reverso.net/german-english/h%C3%A4ufig
No, the particular "usage formula" (so to speak) of nennen that you quoted is not used in the sense you mentioned. The actual meaning is in the sense of mitteilen (to share), in particular to give one or more names to a person. One of the examples Pons gives for it:
Können Sie mir einen guten Anwalt nennen? (Can you give me the name of a good lawyer?)
The sense in which the verb nennen is used in the current exercise is "to name" (benennen). The structure is like this: jemanden/etwas (Akk) etwas (Akk) nennen. Some further examples from DWDS:
er hat mich gestern (laut, vor allen Leuten) einen Dummkopf genannt
das nenne ich Glück
I think you're confusing infinitive of the verb to be, "sein" with the male possessive, "sein."
"Ich bin sein bester Schüler" is nominative. BUT "bin" is the first person singular form of the verb "to be" -- that's where "sein" comes in. The reason it's nominative is that the subject and object are the same thing -- equivalent -- because one of them IS the same as the other.
See "conjugations of the verb to be, sein" at: http://german.about.com/library/verbs/blverb01.htm
Less confusingly, "Ich bin der beste Schüler" is nominative for the same reason.
"Sie hat mich ihren besten Schüler gennant." is accusative, just like "Er hatte mich seinen besten Schüler genannt." The subject, pupil, is receiving the direct action of the verb, to be called.
German word order is different. I think a pronoun object usually comes right after the conjugated verb in a main clause or simple sentence. The time frame usually comes at the beginning of the middle section which would be right after that; then, why and then how and then where.
It does state that a short adverb of time can precede a noun object. The noun object in this case is probably considered to be a part of the ending verb
Then it does go on to say that emphasis would change with different word order, so I suppose it might be possible. Wouldn't it mean that "he had OFTEN called ME his best student. (sometimes he called other people his best student)? Often me, so other times not me? I hope a native speaker comes along to explain how this word order would be interpreted. I think in the regular word order "oft" affects the verb rather than the pronoun.
There are different uses for "nennen" and some use Accusative/Accusative, especially calling someone something. Yes, "name your price" or "state your preference" takes Accusative. http://dictionary.reverso.net/german-english/nennen There is even a context for Accusative/Dative when you name a child after someone. (the preposition Nach takes jdm or Dative) http://german.about.com/od/grammar/ht/Dative-Prepositions.htm