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In Greek mythology, Zeus places his son born by a mortal woman, the infant Heracles, on Hera's breast while she is asleep so the baby will drink her divine milk and become immortal. Hera wakes up while breastfeeding and then realizes she is nursing an unknown baby: she pushes the baby away, some of her milk spills, and it produces the band of light known as the Milky Way. -- Wikipedia

I suspect the band of light looked whiter than the black sky and so was named that way first, and the the Hellenists invented a -- I'm going to lose the proper word here -- explanatory myth that tells the tale of how it got to be that way.

For most of the classical age, Poseidon was the big daddy, and Zeus came later, so I think this myth is late in the game.