Mizraim (Hebrew: מִצְרַיִם / מִצְרָיִם, Modern Mitzráyim Tiberian Miṣrāyim / Miṣráyim ; cf. Arabic مصر, Miṣr) is the Israeli Hebrew and Aramaic name for the land of Egypt, with the dual suffix -āyim, perhaps referring to the "two Egypts": Upper Egypt and Lower Egypt.
Neo-Babylonian texts use the term Mizraim for Egypt. The name was for instance inscribed in the famous Ishtar gate of Babylon. Ugaritic inscriptions refer to Egypt as Msrm, in the Amarna tablets it is called Misri, and Assyrian and Babylonian records called Egypt Mu-ṣur and Muṣri. The Arabic word for Egypt is Miṣr (pronounced Maṣr in Egyptian colloquial Arabic), and Egypt's official name is Gumhuriyyat Miṣr al-‘Arabīyyah (the Arab Republic of Egypt).
According to Genesis 10, Mizraim (a son of Ham) was the younger brother of Cush and elder brother of Phut and Canaan, whose families together made up the Hamite branch of Noah's descendants. Mizraim's sons were Ludim, Anamim, Lehabim, Naphtuhim, Pathrusim, Casluhim (out of whom came Philistim), and Caphtorim.
According to Eusebius' Chronicon, Manetho had suggested that the great age of antiquity in which the later Egyptians boasted had actually preceded the flood, and that they were really descended from Mizraim, who settled there anew. A similar story is related by medieval Islamic historians such as Sibt ibn al-Jawzi, the Egyptian Ibn Abd-el-Hakem, and the Persians al-Tabari and Muhammad Khwandamir, stating that the pyramids, etc. had been built by the wicked races before the deluge, but that Noah's descendant Mizraim (Masar or Mesr) was entrusted with reoccupying the region afterward. The Islamic accounts also make Masar the son of a Bansar or Beisar and grandson of Ham, rather than a direct son of Ham, and add that he lived to the age of 700. Some scholars think it likely that Mizraim is a dual form of the word Misr meaning "land", and was translated literally into Ancient Egyptian as Ta-Wy (the Two Lands) by early pharaohs at Thebes, who later founded the Middle Kingdom.
But according to George Syncellus, the Book of Sothis, supposedly by Manetho, had identified Mizraim with the legendary first Pharaoh Menes, said to have unified the Old Kingdom and built Memphis. Mizraim also seems to correspond to Misor, said in Phoenician mythology to have been father of Taautus who was given Egypt, and later scholars noticed that this also recalls Menes, whose son or successor was said to be Athothis.
In Judaism, Mitzrayim has been connected with the word meitzar (מיצר), meaning "sea strait", possibly alluding to narrow gulfs from both sides of Sinai peninsula. It also can mean "boundaries, limits, restrictions" or "narrow place."
However, author David Rohl has suggested a different interpretation: "Amongst the followers of Meskiagkasher [Cush] was his younger 'brother'– in his own right a strong and charismatic leader of men. He is the head of the falcon tribe – the descendants of Horus the 'Far Distant'. The Bible calls this new Horus-king 'Mizraim' but this name is, in reality, no more than an epithet. It means 'follower of Asr' or 'Asar' (Arabic m-asr with the Egyptian preposition m 'from'). Mizraim is merely m-Izra with the majestic plural ending 'im'. Likewise, that other great Semitic-speaking people – the Assyrians – called the country of the pharaohs 'Musri' (m-Usri)." 
- Brooks, Joshua William (1841), The history of the Hebrew nation: from its origin to the present time, R.B. Seeley and W. Burnside