"Greek nobles whose descent was traced to Cecrops or Eurysthenes or to the great Ionian and Minyan houses"
Public DomainCecrops - Credit: Wikimedia Commons
These are all legendary or semi-mythical characters that Greek nobility might claim to be descended from, in much the same way that families like the Julii in Rome claimed to be descended from legendary Roman heroes. Cecrops was a mythical king of ancient Athens, who judged the contest for the patronage of the city between the goddess Athena and the god Poseidon. Eurysthenes was a great-great-great-grandson of Heracles (Hercules) and the founder of a dynasty of Spartan kings. The Ionians were one of the tribes (along with the Dorians, Achaeans and Aeolians) that the Greeks of the Homeric Age were divided into; the “great Ionian houses” were the houses of the heroes of Homeric legends. The Minyans were one of the mythical indigenous peoples of ancient Greece, older even than the Homeric heroes. The Greeks began to colonise Sicily in the 8th century BC, the most important colony being the city of Syracuse.  

Greek Temple of Segesta, Sicily
Creative Commons AttributionGreek Temple of Segesta, Sicily - Credit: Paul Stephenson/Wikimedia Commons