Phoenix was the son of Agenor and Telephassa. He was also the brother of Cadmus, the father (some say) of Europa, and gave his name to the Phoenicians. Another Phoenix was the son of Amyntor, king of Ormenium, and Cleobule.
The name means 'blood-red' {phoinos), and doubtless referred to the moon. Many of the names of the family of Minos relate to the moon or other heavenly bodies, or could be interpreted this way.
Europa, for example, is 'broad face' and the daughter of Phoenix and Telephassa ('far-shining') as well as being the mother of Minos. Moreover she married Asterion ('starry'). Meanwhile the wife of Minos is Pasiphaë ('all-shining'), who herself is the daughter of Helios (the sun). In turn, Pasiphaë's daughter is Phaedra ('bright').
Neither Phoenix considered here is anything to do with the Phoenix, the fabulous bird of Arabian (or Egyptian) mythology that was said to set fire to itself and rise from the ashes every five hundred years. Tales of this bird were introduced to the Greeks by Herodotus.
Its name also means 'blood-red', 'purple', either with reference to the colour of its plumage (or part of its plumage) or to the flames of the fire on which it perished. The tales and origins of the Phoenix are so complex that they deserve a book of their own.