Aegyptus is the name of two figures in Greek mythology.
Aegyptus, King of Egypt and Arabia 
According to Greek mythology, Aegyptus (Ancient Greek: Αἴγυπτος, Aígyptos) is a descendant of the heifer maiden, Io, and the river-god Nilus, and was a king in Egypt. Aegyptos was the son of Belus and Achiroe, a naiad daughter of Nile. Aegyptus fathered fifty sons, who were all but one murdered by forty nine of the fifty daughters of Aegyptus' twin brother, Danaus, eponym of the Danaids.
A scholium on a line in Euripides, Hecuba 886, reverses these origins, placing the twin brothers at first in Argolis, whence Aegyptus was expelled and fled to the land that was named after him. In the more common version, Aegyptus commanded that his fifty sons marry the fifty Danaides, and Danaus with his daughters fled to Argos, ruled by Pelasgus or by Gelanor, whom Danaus replaced. When Aegyptus and his sons arrived to take the Danaides, Danaus relinquished them, to spare the Argives the pain of a battle; however, he instructed his daughters to kill their husbands on their wedding night. Forty-nine followed through, but one, Hypermnestra ("greatly wooed"), refused, because her husband, Lynceus the "lynx-man", honored her wish to remain a virgin. Danaus was angry with his disobedient daughter and threw her to the Argive courts. Aphrodite intervened and saved her. Lynceus and Hypermnestra founded the lineage of Argive kings, a Danaid Dynasty.
In some versions, Lynceus later slew Danaus as revenge for the death of his brothers, and the Danaides were punished in the underworld by being forced to carry water through a jug with holes, or a sieve, so that the water always leaked out.
Aegyptus is not a genuinely Egyptian figure but a figment of Egypt in the European imagination.
Aegyptos of Thessaly 
In the second or third century AD, Antoninus Liberalis tells of another Aegyptos, who was a young man of Thessaly. He was the companion of Neophron, but the lover of Timandra, Neophron's mother; he became the victim of Neophron's revenge, when Neophron arranged a night-time substitution, so that Aegyptos committed involuntary incest with his mother, Bules. Zeus transformed Aegyptos and Neophron into eagles and Timandra into a kite. Many of the transformations in Antoninus' prose compilation are found nowhere else, and some may simply be inventions of Antoninus; this story combines several themes of Hellenistic romance. The placement of an Aegyptus in Thessaly is inexplicable.
- Egypt took its name from his, according to folk etymology (see the article Copt); thus for Euripides, in his tragedy Helen, Aegyptus has become Egypt itself: "Proteus, while he lived, was King here, ruling the whole of Aigyptos from his palace on the island of Pharos."
- "Belos", "lord", is simply a Hellenized rendition of Baal, a Semitic term, not an Egyptian one.
- According to pseudo-Apollodorus, Bibliotheke 2.1.4-5.
- An eponym for autochthonous peoples, here represented as pre-Hellenic.
- Antoninus Liberalis, Metamorphoses, v.
- Stewart, M. People, Places & Things: Aegyptus (1), Greek Mythology: From the Iliad to the Fall of the Last Tyrant. 
- Jean Vertemont, Dictionnaire des mythologies indo-europeenes, 1997.