Midrash Rabba - Eicha - Petichata - 2 part iv

ר' יוחנן ור' שמעון בן לקיש ורבנן ר' יוחנן אמר למלך שהיו לו שני בנים כעס על הראשון נטל את המקל וחבטו והגלהו אמר אוי לזה מאיזו שלוה נגלה כעס על השני ונטל את המקל וחבטו והגלהו אמר אנא הוא דתרבותי בישא כך גלו עשרת השבטים והתחיל הקב"ה אומר להם את הפסוק הזה (הושע ז')אוי להם כי נדדו ממני וכיון שגלו יהודה ובנימן כביכול אמר הקב"ה (ירמיה י')אוי לי על שברי

Rabbi Yochanan, Rabbi Shimon ben Lakish, and the Rabanan.
Rabbi Yochanan said, {a parable} to a king who has two sons. He became angry with the first one, took the staff and hit him, and exiled him. He said, 'Woe to this one, from what security he was exiled.' He became angry with the second one, and took the staff and hit him, and exiled him. He {the king} said, 'I am he whose situation is grim.'

Similarly, the 10 Tribes were exiled, and Hashem began to say to them this verse (Hoshea 7:13): אוֹי לָהֶם כִּי-נָדְדוּ מִמֶּנִּי - "Woe unto them! For they have strayed from me. And once {the tribes of} Yehuda and Benyamin were exiled, it was as if Hashem said, (Yirmiyahu 10:19) אוֹי לִי עַל-שִׁבְרִי - "Woe is me for my hurt!"

רשב"ל אמר למלך שהיה לו שני בנים כעס על הראשון ונטל את המקל וחבטו ופרפר ומת התחיל מקונן עליו כעס על השני ונטל את המקל וחבטו ופרפר ומת אמר מעתה אין בי כח לקונן עליהם אלא קראו למקוננות ויקוננו עליהם כך גלו עשרת השבטים והתחיל מקונן עליהם (עמוס ה')שמעו את הדבר הזה אשר אנכי נושא עליכם קינה בית ישראל וכיון שגלו יהודה ובנימין כביכול אמר הקב"ה מעתה אין בי כח לקונן עליהם הה"ד (ירמיה ט')קראו למקוננות וגו' ותמהרנה ותשאנה עלינו נהי עליהם אין כתיב כאן אלא עלינו דידי ודידהון ותרדנה עיניהם דמעה אין כתיב כאן אלא עינינו דידי ודידהון ועפעפיהם יזלו מים אין כתיב כאן אלא ועפעפינו דידי ודידהון

Rabbi Shimon ben Lakish said, {a parable} to a king who had two sons. He became angry with the first one and took the staff and hit him, and he spasmed and died. {Etz Yosef says Resh Lakish has the son die in the parable since it is custom to say laments only on the dead.} He began to lament upon him. He became angry with the second one and took his staff and hit him, and he spasmed and died. He said, 'Now I have not within me the strength to lament upon them. Rather, (Yirmiyahu 9:16) קִרְאוּ לַמְקוֹנְנוֹת, "call for the mourning {lamenting} women..."

Similarly, the 10 tribes were exiled, and He began to lament upon them. (Amos 5:1) שִׁמְעוּ אֶת-הַדָּבָר הַזֶּה, אֲשֶׁר אָנֹכִי נֹשֵׂא עֲלֵיכֶם קִינָה--בֵּית יִשְׂרָאֵל - "Hear ye this word which I take up for a lamentation over you, O house of Israel." And once {the tribes of} Yehuda and Binyamin were exiled, it was as if Hashem said, 'Now I do not have the strength to say lamentations.' This is what is written, (Yirmiyahu 9:16) קִרְאוּ לַמְקוֹנְנוֹת, "call for the mourning {lamenting} women..." {though the verse actually has a vav in the beginning - וְקִרְאוּ לַמְקוֹנְנוֹת.)

{
Thus, Resh Lakish contrasts the exile of the 10 tribes, when Amos speaks and says the lament - or perhaps citing Hashem who takes up the lament, with the exile of Binyamin and Yehuda when Yirmiyahu speaks of calling for professional female wailing women to take up the lament.

He continues to interpret from Yirmiyahu, in this case, the next verse, Yirmiyahu 9:17, about what these mourning women will do:

וּתְמַהֵרְנָה, וְתִשֶּׂנָה עָלֵינוּ נֶהִי; וְתֵרַדְנָה עֵינֵינוּ דִּמְעָה, וְעַפְעַפֵּינוּ יִזְּלוּ-מָיִם.
"And let them make haste, and take up a wailing for us, that our eyes may run down with tears, and our eyelids gush out with waters."

}

וּתְמַהֵרְנָה, וְתִשֶּׂנָה עָלֵינוּ נֶהִי - "And let them make haste, and take up a wailing for us" - {Resh Lakish focuses on the fact that the phrase we might expect is "they took up a wailing for them, since after all, it is Hashem who is speaking about the tribes of Yehuda and Binyamin. Since they are not wailing for Hashem, but only those Hashem is speaking about, it should say them, not us.}

"for them" it does not say but rather "for us." Mine {Hashem's} and theirs {Yehuda and Binyamin}.

{וְתֵרַדְנָה עֵינֵינוּ דִּמְעָה}
וְתֵרַדְנָה עֵינֵהם דִּמְעָה - "that their eyes may run down with tears" is not written here, but rather "that our eyes may run down with tears." Mine and theirs.

{וְעַפְעַפֵּינוּ יִזְּלוּ-מָיִם}
וְעַפְעַפֵּיהם יִזְּלוּ-מָיִם - "and their eyelids gush out with waters" is not written here, but rather "and our eyelids gush out with waters." Mine and theirs.

רבנן אמרין למלך שהיו לו שנים עשר בנים ומתו שנים התחיל מתנחם בעשרה מתו עוד שנים והתחיל מתנחם בשמונה מתו שנים התחיל מתנחם בששה מתו שנים התחיל מתנחם בארבעה מתו שנים התחיל מתנחם בשנים וכיון שמתו כולם התחיל מקונן עליהם איכה ישבה בדד:

The Rabanan said, {a parable} to a king who had 12 sons, and two died. He began to take solace in the {remaining} 10. Another two died, and he began to take solace in the {remaining} 8. Another two died, and he began to take solace in the {remaining} 6. Two died, and he began to take solace in the {remaining} 4. Two died, and he began to take solace in the {remaining} 2. And once they all died, he began to lament upon them, (Eicha 1:1), אֵיכָה יָשְׁבָה בָדָד, "How doth sit solitary..."

{It seems that the Rabanan are focusing on בָדָד, solitary. If so, the midrash may be to take this phrase and apply it to Hashem, who now sits alone, where He previously had many sons = tribes. From the same verse, רַבָּתִי עָם - that was full of people. This can be trying in part to address those verses cited by Resh Lakish, where he derived that the lament was both for the nation and Hashem. Otherwise, the purpose of listing the 12 is just to match up to the first verse of Eicha, about how it was once full (with 12 children=tribes) and is now desolate.}