Turns out, it was old Isaac Newton, and his dedication to the ancient Greek ideas. Here is a clip from a blog by Patricia Fara, a senior lecturer at Cambridge, and author of "Science: A Four Thousand Year History" at the Nature web site.
Consider Isaac Newton. He believed so firmly in the Greek idea of a harmonic universe that he divided the rainbow into seven colours to correspond with the musical scale. Before then, although opinions varied, artists mostly showed rainbows with four colours. It is, of course, impossible to make any objective decision about the correct number, because the spectrum of visible light varies continuously: there is no sharp cut-off between bands of different colours, so how you think about a rainbow affects how you see it. Be honest - can you tell the difference between blue, indigo and violet?Dr. Len Fisher has written that, “The mediaeval rainbow had just five colours: red, yellow, green, blue and violet. But Newton added two more – orange and indigo.... The background, though, is that Newton believed that the rainbow should have seven colours, because his view of the harmony of nature required that the colours should be “divided after the manner of a Musical Chord”
I found this on Wikipedia: "n Classical Antiquity, Aristotle had claimed there was a fundamental scale of seven basic colors. In the Renaissance, several artists tried to establish a new sequence of up to seven primary colors from which all other colors could be mixed. In line with this artistic tradition, Newton divided his color circle, which he constructed to explain additive color mixing, into seven colors. His color sequence with the unusual color indigo is still kept alive today by the Roy G. Biv mnemonic. Originally he used only five colors, but later he added orange and indigo, in order to match the number of musical notes in the major scale."
I have found a number of illustrations related to light in which the letters ROYGBIV were spelled out showing the spectrum, such as this clip from The Sunday School Teacher(1867)
One earliest use I have found for the use as a mnemonic, or "aid memorie" as it is called in the article, is from the May 2, 1890 issue of the magazine, Photographic Times:
An even earlier use appears in School managment, by Joseph Landon, in 1883, in a section of the book called Memory in Education :
So I can't find out who first came up with the mnemonic, but I did find a song about it...lyrics are here. And Aaron Wagner sent me a link to a video of a different ROY G BIV song by "They Might Be Giants" that his four-year old highly endorses. Thanks to Aaron and his child for the tip.