In Lamentations, the phrase "daughter of Edom" is repeated in consecutive verses:
Rejoice and be glad, O daughter of Edom, you who dwell in the land of Uz; but to you also the cup shall pass; you shall become drunk and strip yourself bare. (4:21) [ESV]
שִׂישִׂי וְשִׂמְחִי בַּת־אֱדֹום יושֶׂבתי בְּאֶרֶץ עוּץ גַּם־עָלַיִךְ תַּעֲבָר־כֹּוס תִּשְׁכְּרִי וְתִתְעָרִֽי
The punishment of your iniquity, O daughter of Zion, is accomplished; he will keep you in exile no longer; but your iniquity, O daughter of Edom, he will punish; he will uncover your sins. (4:22)
תַּם־עֲוֹנֵךְ בַּת־צִיֹּון לֹא יֹוסִיף לְהַגְלֹותֵךְ פָּקַד עֲוֹנֵךְ בַּת־אֱדֹום גִּלָּה עַל־חַטֹּאתָֽיִךְ
In each case "daughter of Edom" is בַּת־אֱדֹום.
However, the LXX renders "Edom" first as Ιδουμαίας and then as Εδωμ:
Rejoice, and be glad, O daughter Idumea, you who live in the land; indeed the cup of the Lord will pass on to you; you will become drunk and spill. (4:21 NETS)
χαῗρε καὶ εὐφραίνου θύγατερ Ιδουμαίας ἡ κατοικοῦσα ἐπὶ γῆς καί γε ἐπὶ σὲ διελεύσεται τὸ ποτήριον κυρίου καὶ μεθυσθήσῃ καὶ ἀποχεεῗς
Your lawlessness is ended, O daughter of Sion; he will not add to exile you; he visited your lawlessness, O daughter of Edom; he uncovered your impious deeds. (4:22 NETS)
ἐξέλιπεν ἡ ἀνομία σου θύγατερ Σιων οὐ προσθήσει ἔτι ἀποικίσαι σε ἐπεσκέψατο ἀνομίας σου θύγατερ Εδωμ ἀπεκάλυψεν ἐπὶ τὰ ἀσεβήματά σου
What is the significance of using Ιδουμαίας and then Εδωμ to identify the daughter of Edom?