On the second day of Rosh Hashana, after reading about the akeda (and some other stuff), we read Yirm'ya 31:1–19 as the haftara: why this haftara?
The Haftorah is very moving and probably hold the record for the most Jewish songs from one Haftorah! It especially relates to the Rosh Hashanah theme of Zichronos ("Memories").
First, (verses 1-13), the Haftorah discusses God bringing the redemption, which may connect to the theme of Zichronos, as it involves God "remembering" the Jews. Next (verses 14-16) is the scene of Rachel weeping and God promising redemption, which is part of the theme of Zichronos - recalling the patriarchs or matriarchs as a merit to end the exile.
Then (verses 17-18) describe Ephraim's teshuvah (repentance), including the different parts of teshuvah, an important thing to mention during the Aseres Yemei Teshuvah (Ten Days of Repentance):
כִּי אַחֲרֵי שׁוּבִי נִחַמְתִּי וְאַחֲרֵי הִוָּדְעִי סָפַקְתִּי עַל יָרֵךְ, בֹּשְׁתִּי וְגַם נִכְלַמְתִּי כִּי נָשָׂאתִי חֶרְפַּת נְעוּרָי
Surely after that I was turned, I repented, and after that I was instructed, I smote upon my thigh; I was ashamed, yea, even confounded, because I did bear the reproach of my youth.
Finally, as Rashi on Megilla 31a (s.v. "הבן יקיר לי אפרים") points out, verse 19 cites God remembering Ephraim and having mercy on him, which, appropriately enough, is one of the verses of Zichronos included in the Rosh Hashanah mussaf.