The Hebrew name for Egypt, misrayim, corresponds to Ugar. msrm, Phoen. msrym, Egyptian Aram. msryn, Syr. mesrem, Akk. Musur/Musru/Misri, Old Persian Mudraya, Arab. Misr; the word is not, however, attested in Old Egyptian. The Egyptians themselves called their land ... t3.wy, the "two lands" (referring to Upper and Lower Egypt) ... If misrayim constitutes a genuine dual form, and if it is connected with Akk. misru, "border, region", and Arab. misr, "border, land, capital city," it might be a translation of t3.wy, although this explanation is extremely uncertain. Meir Fraenkel's derivation of misrayim, associating it with matar, "rain," "water," is untenable.A few notes about this entry:
The singular form masor also occurs in the OT (Mic. 7:12; 2 . K 192:24 par Isa. 37:25; Isa. 19:6) and the gentilic form misri is richly attested.
1) There is a Hebrew cognate to the Akkadian misru -metzer מצר, which also means "boundary". According to Klein, the word only appears in post-biblical Hebrew (e.g. Bava Batra 61b, 62b) and is a secondary form of the biblical word metzar מצר. That word is said to mean either "distress" or "a narrow place" ("strait" in modern Hebrew). It is familiar from the phrase בין המצרים bein hametzarim - which is used to describe the three weeks preceeding Tisha B'Av. The origin of the phrase is Eicha 1:3 -
כָּל-רֹדְפֶיהָ הִשִּׂיגוּהָ, בֵּין הַמְּצָרִיםThe JPS translates this as "all her pursuers overtook her in the narrow places", but adds their common note to "in the narrow places": "Meaning of Hebrew uncertain". The Daat Mikra says that "narrow places" is probably the plain meaning of the verse, but says there are those that explain it as meaning "distress", so the verse would mean "all her pursuers overtook her when she was in distress." Interestingly, they also quote the Rashbam as saying the word here means "border" - in the borders of the Kingdom of Yehuda, and that whenever the Jews would flee to the border, their neighbors would hand them over to their enemies. So perhaps the word does have a biblical origin.
2) The Arabic misr meaning "capital city" explains how Cairo is referred to as Masr in modern Egyptian Arabic. Stahl, in his Arabic etymological dictionary, writes that this title originally applied to Fustat. He quotes Maimonides, in a letter to Ibn Tibbon, where he writes, "I reside in Mitzrayim [meaning Fostat]; the king resides in Cairo."
3) The Daat Mikra generally writes that Matzor מצור is simply a poetic form of Mitzrayim, and not a singular form (however in their commentary to Bereshit 10:6, when the word Mitzrayim first appears in the Tanach, they write that it might be a double form, indicating the two kingdoms.) In a footnote to Melachim II 19:25, they note that Matzor might preserve an earlier form of the name, and that the suffix "-im" in Mitzrayim might be locative, like Yerushalayim. This explains the Akkadian and Arabic forms, which otherwise might appear to be singular.
4) Steinberg, while clearly aware of the division of Egypt into the Upper and Lower Kingdoms (he mentions it in his entry), writes that the plural nature of Mitzrayim is due to the Nile river splitting the country into east and west (or as this book writes, "the two banks of the Nile"). This could help answer the question raised in this book, who after acknowledging the theory that Mitzrayim is a dual form, writes:
However, prophetic texts from Jeremiah [44:1] and Isaiah [11:11] differentiate between מצרים [Mitzrayim] and פרתס [Partos] as Lower and Upper Egypt, indicating that מצרים [Mitzrayim], if it is to be located as a geographic reference, at least in these prophetic texts refers to Lower Egypt or the Nile delta.This site writes that:
Northern or Lower Egypt is called Mazor, .. while Southern or Upper Egypt is Pathros, the Egyptian Pa-to-Res, or "the land of the south" (Isa. 11:11). But the whole country is generally mentioned under the dual name of Mizraim, "the two Mazors".However, this explanation ignores the fact that Mitzrayim and Patros are listed together in Yishayahu and Yirmiyahu.
One unexpected derivative of Mitzrayim - actually the Arabic Misr - is the song Misirlou. Even if you don't recognize the name (it means "Egyptian girl"), there's a good chance you know the song. Listen to this brief NPR story, and you'll hear how the song spread all over the world, including to the niggunim of rabbis and to klezmer bands. I guess you can take the Jews out of Mitzrayim, but you can't really take Mitzrayim out of the Jews...