I started off by going to Wikipedia and seeing what sources they had. They had this quote: "Raine, raine, goe to Spain: faire weather come again"and credited James Howell.
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Then it was time to search for James Howell's book of proverbs dated in 1659. I found a text typed and searchable from the University of Michigan, which helped find the text in a really poor scan found at Archive.org. Both of these searches were hampered by the fact that the original quote spells "rain" without the e.
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The next text mentioned was a book titled "Oranges and Lemons: Rhymes from Past Times" by Karen Dolby, published in 2012. This book looks like a great source of material, but seems to not have a definitive source for the Ancient Roman/Greek claim found on Wikipedia.
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The next text mentioned is Poor Humphrey's Calendar (1829) which you can see looks like an amazing collection of foretellings. Apparently, it was all the rage to have publications by characters rather than people, which is why Benjamin Franklin had created and titled Poor Richard's Almanack.
Sent from Wikipedia: Andrew Cheviot's Proverbs, proverbial expressions, and popular rhymes of Scotland; collected and arr., with introd., notes and parallel phrases, 1896
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The selection from the left is from The Nursery Rhymes of England : Obtained Principally from Oral Tradition (1842) by James Halliwell
This page is from what Wikipedia calls "Early English Poetry, Ballads, and Popular Literature of the Middle Ages" (Percy Society, 1846), but what I found to be called A Collection of Proverbs and Popular Sayings Relating to the Seasons, the Weather, and Agricultural Pursuits, Vol. 20, 1846 by the same author, Michael Aislabie Denham. He was part of the Percy Society, which was a group that wished to publish rare poems and songs.
The Penguin Guide to the Superstitions of Britain and Ireland by Stephen Roud (the guy who created the Roud index number for folksongs!)
The first sources is listed as "Unlocated Aubrey," which apparently means they are fairly sure John Aubrey wrote it in 1686, but they don't know where.
The Fulcher Collection is in the The Folklore Society and I am not a member and could not access.
The "Lincolnshire Rudkin" (Lincolnshire Folklore by Ethel Rudkin) was not available on Google Books or Archive, however I was able to search the text on Google Books and was unable to find "Rain, Rain" in the text.
The "Denham Tracts" turned into "The Denham tracts; a collection of folklore, reprinted from the original tracts and pamphlets printed by Denham between 1846 and 1859" by Michael Aislabie Denham and James Hardy. A search of the text resulted in not finding the words, but we may not be working from the complete collection.
"Henderson (1866)" can be found below.
I was unable to get through paywalls to access the "Newcastle Weekly Chronicle (11 Feb. 1899).
I found the "Yorkshire Addy" materials published below.
The "Halliwell" 1849 is located below.
William Henderson's Notes on the folk-lore of the northern counties of England and the borders - the version I found was published in 1879, but the Penguin Guide said it would be published in 1866.
Sidney Oldall Addy mentions the poem on p. 188 of Household Tales with other Traditional Remains, mentions the text as collected from East Riding of Yorkshire.
James Halliwell's sequel to the above-cited tome, Popular Rhymes and Nursery Tales: a sequel to the Nursery Rhymes of England,was published in 1849 and has a little more context and variation in the rhyme.
The Roud sent me to A Dictionary of Superstitions by Iona Opie and Moira Tatum, which mentioned many sources we have already seen.
The Opie text sent me to another Opie text, The Lore and Language of School Children, by Peter Opie and Iona Opie, published in 1987.
Then, of course, I had to go looking for versions that had the addition of "Little __________ wants to play."
The Book of Nursery Rhymes, Tales, and Fables: A Gift for All Seasons from 1847, edited by Lawrence Lovechild
Old Nurse's Book of Rhymes, Jingles and Ditties from 1858, p. 40, edited by Charles H. Bennett.
The first melody was from The Illustrated Book of Nursery Rhymes and Songs, with Music. Edited by T. L. Hately. Illustrations by Keely Halswelle from 1865