Why did she say "Bi Adoni
Rashi: Eli should oversee him, that Shmuel will be his Talmid. Radak, from Targum Yonasan - Bi Adoni is an expression of supplication and request.
Rashi citing Brachos 31b: Eli should not kill Shmuel for teaching a Halachah in front of his Rebbi.
What is the meaning of "Imechah"?
Radak #1: [I prayed] in front of you, in your presence.
Why does "Imechah" have a Hei?
Radak #1: Sometimes Chaf Hei is at the end of a word in place of final Chaf, e.g. "Al Yadecha" (Shemos 13:16).
Radak citing Brachos 31b: [Eli was standing because] one may not sit within four Amos of someone praying. Tosfos (31b) - the Hei hints that he may sit in the fifth Amah [from the one praying].
Why did she say "ba'Zeh"?
Malbim: She gave reasons why Eli should accept to raise Shmuel. (a) I prayed for him in your presence, and you helped me with your prayer. Ba'Zeh - I prayed in this corner. And I prayed El Hashem, and not via intermediaries. Since he was born [partially] due to the Kohen's prayer, it is proper that he serve the Kohen in this place. (b) I am the same woman who prayed here. My body did not change. This was special Hashgachah that I, who was barren, gave birth. This shows that the child was prepared to be Ish Elokim from the womb. Refer also to 1:27:1:2 for more than reasons.
