The Malbim contends that Esther, besides possessing health and beauty, also had great character because of her distinguished father. We know that he was a great man because it says in the Talmud (hinted at in Megillah 10b) that all prophets must have good genealogy.
Another reason for her father to be mentioned here comes from the Maharal. He quotes the verse in the Torah (Bireishis 2:24) that says a man who finds his intended should cling to her. Maharal continues that a woman, too, clings to her husband after marriage. Therefore, Esther was connected to Mordechai up until this point, and will now have to cling to her new “husband,” Achashverosh.
The Talmud (Megillah 13b) says that Esther’s ancestors Rachel, Benyamin, and Shaul were all able to keep secrets. This characteristic was passed down through Avichayil to Esther. Rav Chaim Kanievsky says this verse emphasizes this genetic link to secrecy. This is why Esther’s father was not mentioned earlier when her secrecy was first mentioned (2:10 above), because there, she was commanded to be secretive by Mordechai, and this verse is attesting to her innate ability to do so for this long period of time.