Why does it say "Medaberes Al Libah"?
Brachos 31b: She discussed matters of (on) her heart. She said, You did not create anything in a woman for naught; there are organs to see, to hear..., breasts to nurse -- give me a child, so I can use mine!'
Radak citing Targum Yonasan: She prayed in her heart. She had intent in her prayer. 1
Bereishis Rabah 34:10: This teaches that Tzadikim 2 , their hearts are in their Reshus (control). Resha'im are in their heart's Reshus - "Amar Naval b'Libo" (Tehilim 53:2), and regarding Esav, Yaravam and Haman (Bereishis 27:41, Melachim 1 12:26, Esther 6:6)
Yalkut Shimoni (78): Why are women joined with minors and slaves for Mitzvos 3 ? It is because they have only one heart 4 ; we learn from Libah. (and not Levavah, similar to Levavchem).
Malbim: She prayed quietly.
Brachos 31b: We learn from her that one who prays must have intent. Also, his lips must articulate the words, but his voice may not be heard - "Sefaseha Na'os v'Kolah Lo Yishame'a."
Avraham said b' Libo (Bereishis 17:17)! Refer to Bereishis 17:17:4:1,2.
Zayis Ra'anan: They are exempt from Mitzvas Aseh sheha'Zman Gerama. (Minors are exempt from all Mitzvos! According to Mei ha'Chesed (refer to 1:13:1:4**), perhaps 'they are joined', i.e. a man may not be Yotzei a Mitzvah through them. Since Daitan Kalos, they cannot intend both for themselves and for others. However, they can be Motzi (a) others like themselves, and (b) even men, when they are equally obligated! This is difficult. The entire Drashah is difficult to me, for Pesukim refer to the heart in the singular also for men, e.g. David and Avraham (27:1, Bereishis 17:17). It says Levavah about Malkas Sheva (Melachim 1 10:2), and some say that this was a woman (Ibn Ezra Daniel 11:6, Maharsha Bava Basra 15b) - PF)
Zayis Ra'anan: I.e. the Yetzer Tov does not rule over them so much. If a Mitzvah does not apply always, they are prone not to do it. (This is astounding, especially since the verse discusses Chanah, an awesome Tzadekes! - PF) Therefore, the Torah exempted them. Mei ha'Chesed (on the Midrash) - this is unlike "v'Sa'adu Libchem" (Bereishis 18:5). Angels have no Yetzer ha'Ra! Rather, Daitan Kalos; their mind is only on one matter (intimacy with her husband, childish follies, and his master), to exclude everything else.
Why did he think that she was drunk because her voice was not heard?
Rashi: People did not used to pray quietly. Malbim - sometimes people would pray to Hashem, and sometimes to angels or other advocates. (a) When one prays to Hashem, he need not elaborate, for he does not justify [his request based on] his merits; he requests Chesed and a free gift. When one prays via advocates, he recounts his merits to clarify that he deserves his request. 1 (b) When one prays to Hashem, he should not be audible, this shows lack of Emunah. Hashem knows all thoughts! When one prays via advocates, he can 2 raise his voice, for angels do not know [people's] thoughts. In the days of Eli people used to ask via advocates, for they were not worthy of miracles and special Hashgachah. Therefore they used to elaborate in prayer. Eli assumed that she was doing so, for she elaborated. Therefore, he was Shomer Es Piha (wanted to hear what she said). However, she was not audible. Therefore, he thought that she was drunk.
Radak (14): Others who pray, they say their requests audibly. Even though sometimes one prays quietly, he screams some of his words. You prayed so long, and were not heard [at all].
Malbim: Human kingship is the same; one who requests directly from the king simply requests Chesed; one who requests from the officers who sit in judgment must justify that he is entitled according to law.
Based on his words below, it seems that he means that he must pray audibly, and he may pray loudly if he wants. (PF)
What was the consequence of Eli mistakenly suspecting her?
Ba'al ha'Turim, citing the Rosh: It says here "va'Yachsheveha", like regarding Tamar ("va'Yachsheveha l'Zonah" and "va'Yachsheveha Lo Tzedakah" (Bereishis 38:15, 15:6). One who is wrongly suspected, it is considered Tzedakah for him. In this merit, Shmuel came from Chanah, and kings and prophets from Tamar.