Eurynome was born from Oceanus and Tethys, the primordial deities whose waters encircled the world and whose children numbered three thousand rivers and as many Oceanid nymphs. In the genealogy of Hesiod, the Boeotian poet who composed the Theogony around 700 BCE, she is listed among the elder Oceanids, the daughters of the world-river who were themselves goddesses of some power. But Eurynome was more than a nymph. In the Orphic theogonies, the mystical poems that told a different creation story from Hesiod's, she was a Titan of the first rank, a cosmic dancer who moved upon the face of the waters before the world had shape, and who, with her consort Ophion, the great serpent, ruled Olympus in the age before the Olympian gods were born.

The story of Eurynome and Ophion comes from Pherecydes of Syros, a philosopher and mythographer of the sixth century BCE whose lost work on the gods survives only in fragments quoted by later writers. In Pherecydes's account, Eurynome and Ophion were the first rulers of Olympus, reigning in a golden age before the birth of Cronus and the other Titans. But their reign did not last. Cronus, the son of Uranus and Gaia, rose against them, just as he would later rise against his own father, and cast Ophion and Eurynome down from Olympus into the stream of Oceanus. Some accounts say that Eurynome accepted her fall with grace, dancing upon the waves of her father's river and transforming her defeat into an eternal dance of creation. Others say that she and Ophion dwelt thereafter in the depths of the earth, in the caverns beneath Olympus, waiting for the day when the old order would return. Whatever the truth, her overthrow marked the end of the first dynasty of heaven and the beginning of the Titan age that would itself be overthrown by Zeus.

Hesiod gives Eurynome a gentler role. In the Theogony, she is not a fallen queen but a beloved of Zeus, the king of the gods who succeeded Cronus. She lay with Zeus in a meadow of flowers, and from their union were born the three Charites, the goddesses who personify the pleasures of human life. Aglaea, whose name means Splendour, is the grace of radiant beauty, the glow that surrounds those whom the gods favour. Euphrosyne, whose name means Joy, is the grace of mirth and good cheer, the laughter that binds friends together at the feast. Thalia, whose name means Festivity, is the grace of abundance and flowering, the bloom of youth and the harvest of age. Together they attended the great goddesses of Olympus, weaving garlands for Aphrodite, the goddess of love and desire, and singing hymns for Hera, the queen of the gods. They were present at every wedding, every birth, every festival, and their absence was the surest sign of divine displeasure.

Eurynome's cult was small but ancient. At Phigalia, a city in Arcadia in the central Peloponnese, the traveller Pausanias saw a sanctuary of Eurynome beside the river Neda in the second century CE. The sanctuary contained a wooden statue of the goddess, bound with golden chains, which the locals said was extremely ancient and had been carved by a craftsman whose name was lost in time. The Phigalians told Pausanias that Eurynome was a daughter of Oceanus and the mother of the Charites, and that she had been worshipped at this spot since before the Trojan War. They also said that she was the same goddess whom the people of Homer knew as a daughter of Oceanus who helped Hephaestus, the smith-god, when he was thrown from Olympus by Zeus. In the Iliad, when Hephaestus falls from heaven and lands in the sea, it is Eurynome and Thetis, the sea-goddess, who rescue him and nurse him in their underwater cave, teaching him the craft of metalworking that would make him the most skilled artisan among the gods.

The memory of Eurynome as a cosmic dancer survived in the Orphic tradition, where she was sometimes identified with the goddess who created the world by dancing upon the face of the waters, her movements generating the patterns of creation. This image, of a goddess whose dance is the act of making, connects her to the oldest stratum of Greek religion, the belief that the world was not made by a craftsman or a warrior but by a dancer whose body was the measure of all things. Eurynome's name, which means "wide-ruling" or "of broad pastures," suggests a goddess of abundance and space, a deity whose domain was not a single city or mountain but the whole circumference of the world-encircling river.