Robin started out as a medieval diminutive of Robert. It came to be used for girls in the late 1800s probably following the trend of giving nature names to girls (this is where we get a lot of names like Ivy, Bryony, Pearl etc.) and thus it was in reference to the red-breasted bird rather than the connection to Robert. It caught on as a girls' name in the mid 20th century and now in the USA is used almost exclusively for girls.
See Also: Robert

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