Zeus, and that when swimming through a section of the Adriatic Sea she gave her name to this as the Ionian Sea. (Later, still as a heifer and still swimming, she passed through the Bosporus, which as 'Oxford' - bos, 'ox' and poros, 'ford' - was also named after her.)
She was originally the daughter of Inachus, first king of Argos, and Melia. Traditionally she is associated with the moon, which in a dialect of Argos was called Io, from io, eimi, 'to go'. Equally, she could have been regarded as the feminine equivalent of Ion, and thus as a forerunner of the Ionian race.
Somehow the 'cow and moon' association seems the most attractive, since the inhabitants of Argos are said to have worshipped the moon as a cow, regarding the horned new moon as a source of water, and so of cattle food. There are those, even, who see the cow jumping over the moon in the 'Hey diddle diddle' nursery rhyme as symbolic of the changing seasons and the passing of the months.
Iona and Peter Opie firmly discount such explanations in The Oxford Dictionary of Nursery Rhymes. (It is simply an agreeable coincidence that the female half of this husband-and-wife team has a name that itself may well link up with that of Io!)