I don't believe that Amir reveals his father's real name during any part of The Kite Runner. Nor does Amir ever state his own last name during the text. Amir simply refers to him as "Baba," which means "father" in Arabic. It was the first word that Amir spoke as a baby. Baba is also referred to as Toophan agha ("Mr. Hurricane"), a nickname given to him by his best friend, Rahim Khan. However, we do get a clue to the family name in Chapter 20 when Amir meets an old beggar on the streets of Kabul after his return to Taliban-held Afghanistan. When Amir hears him say that he used to teach at the university, Amir tells him that his mother used to teach there.
"And what was her name?"
Therefore, assuming that Amir's mother did not use her maiden name after she was married (unlikely in the Arab world) while teaching at the university, we can guess that Amir's--and Baba's--last name was Akrami.