Rabbi Akiba Eger (1761-1837), in his notes to the Shulhan Aruch (Orah Haim 186), addresses the case of a boy who eats a meal with bread, and then recites Birkat Ha’mazon, at the very end of the final day before his 13th birthday, just before he becomes a Bar-Misva. If he still experiences satiation after dark, Rabbi Akiba Eger asks, is this boy now required to repeat Birkat Ha’mazon? The reason to require him to repeat Birkat Ha’mazon is because when he recited it, he was obligated in this Misva only by force of the Rabbinic requirement of Hinuch (training in Misvot). Once night falls, and he becomes a Bar-Misva, his experience of satiation seemingly imposes upon him the Biblical obligation of "Ve’achalta Ve’sabata U’berachta" ("You will eat, you will be satiated, and you shall bless" – Debarim 8:10). Since he had recited Birkat Ha’mazon before he was required by force of the Biblical obligation, one might argue that now, when he is required by force of the Biblical obligation, he must repeat Birkat Ha’mazon.
Rabbi Akiba Eger discusses the various sides of this question, and leaves it unresolved.
The Meshib Dabar (Rav Naftali Sevi Yehuda Berlin of Volozhin, 1816-1893) advanced an argument to prove that the boy in this case does not need to repeat Birkat Ha’mazon. The Rama (Rav Moshe Isserles of Cracow, 1530-1572), in discussing the laws of prayer (Orah Haim 53), writes that if a boy becomes Bar-Misva on a Friday night, and the congregation prays Arbit before sunset (as many communities do in the summer), then this boy should not be the Hazzan for Arbit. The fact that the congregation begins Shabbat early, the Rama explains, does not mean that this boy becomes a Bar-Misva at that point, before nightfall on his 13th birthday. Therefore, he is not qualified to serve as Hazzan. The Meshib Dabar notes that the Rama does not seem to have a problem with the boy reciting Arbit before sundown, fulfilling his obligation to recite the Arbit prayer. It was clear to the Rama that the boy fulfills his Arbit obligation that night even though he recites the prayer before becoming a Bar-Misva; the problem is only his eligibility to serve as Hazzan. By the same token, the Meshib Dabar writes, a boy who recites Birkat Ha’mazon after eating just before dark on the evening of his 13th birthday fulfills his obligation even after he becomes a Bar-Misva when night falls.
Hacham Ovadia Yosef, in Yabia Omer, refutes this proof suggested by the Meshib Dabar. He notes that unlike the Misva of Birkat Ha’mazon, the obligation to recite Arbit is not a Biblical requirement; it was enacted by the Sages. Hence, when a boy recites Arbit before sundown on the evening of his 13th birthday, he recites Arbit as a Rabbinic obligation (due to the requirement of Hinuch) – just as he would even after becoming a Bar-Misva at nightfall, since Arbit is a Rabbinic obligation. And for this reason, he fulfills the obligation by reciting Arbit even before nightfall. This is quite different from the case discussed by Rabbi Akiba Eiger, which involves the Biblical obligation of Birkat Ha’mazon. In this case, one could argue that the boy’s recitation of Birkat Ha’mazon before nightfall, which fulfilled his Rabbinic obligation of Hinuch, does not suffice for the Biblical obligation which takes effect at nightfall.
As for the final Halacha, Hacham David Yosef writes in his Halacha Berura (Siman 186) that since Rabbi Akiba Eiger’s question has not been conclusively answered one way or another, there is no clear-cut Halachic ruling. As such, if a boy ate a meal and recited Birkat Ha’mazon just before nightfall on the evening of his 13th birthday, and he still feels satiated after nightfall, he should preferably eat another Ke’zayit of bread and then recite Birkat Ha’mazon.
Summary: If a boy ate a meal and recited Birkat Ha’mazon at the very end of the day before his 13th birthday, and after nightfall, when he becomes a Bar-Misva, he still feels satiated, it is possible that he must repeat Birkat Ha’mazon. Therefore, in such a case, the boy should eat another Ke’zayit of bread and repeat Birkat Ha’mazon.