Way back when Zeus was but a childgod, she, nymph-goddess, and Ophion, snake god, were the first residents of Olympus. Their union brought about all things. Then, along came Kronos and Rheia with their eviction orders, sending her back to the sea from which she came and he to the underworld.
Later, when Zeus took control and he and Hera threw Hephaistos, blacksmith of the gods, off the mountain, Eurynome had been there, done that. Together with Thetis, fellow nymph-goddess, she picked him up, nursed him back to health and sent him back up the mountain.