(Γενέσιος), that is, " the father," a surname of Poseidon, under which he had a sanctuary near Lerna, on the sea-coast. (Paus. 2.38.4.) The name is identical in meaning with Genethlius (γενέθλιος), under which the same god had a sanctuary at Sparta. (Paus. 3.15.7.)
[L.S]A Dictionary of Greek and Roman biography and mythology
Smith, William
A Dictionary of Greek and Roman biography and mythology. William Smith, LLD, ed. 1890