What exactly does Ben-Oni mean?
Rashi, Targum Onkelos and Targum Yonasan: It means 'the son of my pain'. 1
Rashi (Bamidbar 26:13,24): 'The son of my mourning. This was fulfilled partially after Aharon died and (and Bnei Yisrael wanted to return to Egypt, and Levi fought them, and) five families (from the 10 sons of Binyamin) were wiped out. It was fulfilled totally after Pilegesh b'Giv'ah (only 600 men from Binyamin remained).
Moshav Zekenim, citing R. Yeshayah: It is a Nefel (stillborn). Since she was about to die, she thought that he is a Nefel.
Malbim, Ha'Emek Davar: It is my strength. I gave my strength to him; I died through him.
As in Devarim 26:14.
Why did Yakov call him 'Binyamin'?
Rashi #1: Because he was the only one of his sons to be born in Eretz Kena'an, which is south of Aram Naharayim, 1 and Binyamin is the acronym of Ben Yamin (son of the south). 2
Rashi #2 and Rashbam: It means 'the son of my days', 3 since he was born to Yakov in his old age.
Ramban: 'The son of my strength. 4 Malbim
Rashi: And Yamin means south, as in Tehilim, 89:13. See Ramban (16,18), who cites a number of proofs that Aram Naharayim was (not north, but) east of Eretz Kena'an. He concedes however, that, assuming that Rachel died in the vicinity of Efrat (which is in the portion of Yehudah), then he was born in the south of Eretz Kena'an, though not if she died in the area of Har Efrayim, or even if she died in Binyamin's territory, both of which are described as being in the north. Refer to 35:16:2:1*. Harchev Davar (32:3) brings many supports for Rashi.
Rashi: This in turn, explains why "Yamin" is written Malei (with a 'Yud before the 'Nun'. In most places, it is written Chaser, without the 'Yud'). Ha'Emek Davar
Why does it say "when her soul left, for she died"?
Moshav Zekenim: When Rachel wanted to die, they screamed at her. Her Neshamah was bewildered and returned to her, like we still do nowadays, and she called him Ben Oni.
Why did Yakov change from the name that Rachel gave? It is a Mitzvah to fulfill the words of one who died!
Moshav Zekenim #1: When Yakov was old, he leaned on Binyamin. 1
Moshav Zekenim #2: Her name was mistaken (she thought that he is a Nefel - refer to 35:18:1:3, 35:18:2:6). 2
Refer to 35:18:2:3*.
How does this answer his question? Perhaps he called him Ben Oni until his old age; this fulfilled Rachel's words. His question is not clear to me. The Chiyuv to fulfill Divrei ha'Mes is only for property handed over to a third party (CM 252:2)! (PF)
The Nasi of Binyamin was Avidan (Bamidbar 1:11) - Avi Dan (my father judged correctly that I am viable, unlike my mother thought).
