And I think, when it says "apayim" in plural, that Elkanah had two sorrows: 

- The first is that, because he loved Channah and God had closed up her womb, he was anxious and sad for himself about the absence of sons [from her] 

- And the second is that he was upset because her rival, Penina, would provoke Channah to anger about this. And since Penina would provoke Channah, Elkanah had additional sorrows, and therefore it is appropriate to interpret the verse, "And with regard to Channah he gave a portion -- with double the sorrow". 

Talmud Bavli – Tractate Bava Batra 16a.

Rabbi Levi said: The Satan [of Job] and Peninah had noble intentions (leshem shamayim).

Pesikta Rabbati (Ish Shalom) Parasha 43

It says later, "And the L-rd remembered Channah…" (Shm. I 2:21), and it said earlier, "And her rival provoked her bitterly…" (Shm. I 1:6). Peninah would provoke her and irritate her day after day. What would she do?

Rabbi Nahman son of Abba said: Peninah would rise early and say to Channah, "Aren't you going to get up and wash your children's faces so they can go to school?" At noon, she would say, "Aren't you preparing to welcome your children home from school?"…

Rabbi Tanhuma son of Abba said: When they sat down to eat, Elkanah would give everyone of his sons his portion. What would Peninah do in order to provoke Channah? She would say to Elkanah, "Give this son of mine his portion. You did not give that son of mine his portion."

Rashi on Talmud Bavli, Bava Batra 16a

"In order to provoke [herima] her…" So that she would thunder [shetitra'em] towards the Heavens that she was barren and pray.

Abravanel on Shmuel I 1

…And the Raivad interpreted that she provoked her so that she would pray and thunder to God and ask for mercy, for since her husband loved her she had despaired of asking for mercy.

Pesikta Rabbati 43 - For the LORD Remembered Channah
(Midrash Aggada pertaining to the holidays and haftarot of special Shabbatot & chagim)
 

The midrash learns from the triple usage of “why” that her husband tried to console her with the stories of the three barren Matriarchs. He said to her:

“Would Sarah, who was childless, sit and cry all the day?" 

"Would Rebekah, who was barren, not eat?"

"Would Rachel, who was childless, sit by herself, depressed?”.

With these examples, Elkanah tried to infuse Hannah with hope and cause her to eat of the sacrifice and stop crying.

Malbim on Shmuel I 1 (introduction)

"So Channah rose up…" (Shm. I 1:9) -- until now she had relied solely upon her husband, who was a righteous man, to pray for her, and now that she saw that he had given up on compassion [from G-d], she was aroused to prayer for herself…

Talmud Bavli - Tractate Brachot 31a.

R. Elazar said, "Channah flung her words upward," as it is said, "She prayed to or over [al] the L-rd," (Shm. I 1:10) teaching that she flung her words upward.

Talmud Bavli - Tractate Brachot 31b.

And she vowed a vow and said, O L-rd of Zevaot [Hosts] .(Shm. I 1:11) R. Eleazar said: From the day that God created His world there was no man who called the Holy One, blessed be He, Tzeva'ot [hosts] until Channah came and called Him "Tzeva'ot".

Said Channah before the Holy One, Blessed be He: Sovereign of the Universe, of all the hosts and hosts that You have created in Your world, is it so hard in Your eyes to give me one son? A parable: To what does this compare? To a king who made a feast for his servants, and a poor man came and stood by the door and said to them, Give me a bite, and no one took any notice of him, so he forced his way into the presence of the king and said to him, Your Majesty, out of all the feast which thou hast made, is it so hard in thine eyes to give me one piece?

Talmud Bavli - Tractate Megilla 14a.

‘Seven prophetesses’. Who were these? — Sarah, Miriam, Devorah, Channah, Avigail, Chuldah and Esther

Targum Yonatan on Shmuel I 2

"And Channah prayed…" (Shm. I 2:1) with prophetic inspiration. She said, "My son Shmuel is destined to be a prophet in Israel, and in his days Israel will be miraculously saved from the Philistines…"

Talmud Yerushalmi, Berachot, 4:1

Said R. Yose bar Haninah, "From this verse [1 Sam. 13] you learn four things.

(a) Hannah was speaking in her heart: from this you learn that Prayer requires [focus and] concentration.
(b) Only her lips moved: from this you learn that one must mouth the Prayer with one's lips.
(c) And her voice was not heard: from this you learn that one may not raise his voice and pray [rather pronounce each word clearly, but not too loudly].
(d) And Eli took her to be a drunken woman: from this you learn a drunken person is forbidden to pray".

Tosfot, Tractate Rosh Hashana, 16b

Hiyya said: "I never concentrated during prayer in all my days. Once I wanted to concentrate, but I thought about who will meet the king first the Arkafta or the Exilarch. Shemuel said: "I count clouds."

Bun bar Hiyya said: "I count the layers of stone in the wall."

Rabbi Matyana said: "I am grateful to my head, because it bows by itself when I reach 'Modim"