Eddie Rabbit - Drivin’ My Life Away
Adel - Set fire to the rain.
Rain Song – The Youngbloods
“Bug Rain” by Looper (about rain, not bugs)
Enya, “A Day without Rain” (unfortunately, no lyrics).
Chad and Jeremy, “A Summer Song”:
And when the rain
beats against my windowpane
I’ll think of summer days again
and dream of you.
… And that’s when you know it’s really over. 
Happy 60th, Gina!
Neil Innes - Protest Song
It’s Raining – Rick Derringer
Stormy Weather, written by Harold Arlen and Ted Koehler.
First recorded by Ethel Waters in 1933, and covered by nearly everyone in the big band era.
Walking in the Rain – Flash and the Pan
“Summer Rain” Johnny Rivers (yes)
“SummerRain” Belinda Carlisle
Davy Jones. Rainy Jane
Melanie. (Lay Down) Candles in the Rain
Tanya Tucker. The Rainmaker
nursery song: The Eensie Beensie Spider Came Up the Waterspout
Let It Rain - Eric Clapton
Come In From the Rain - Captain & Tennille
Rain Is Falling - E. L. O.
Rain On the Scarecrow - John Mellencamp
Drivin’ Rain - Gov’t Mule
Laughter In the Rain - Neil Sedaka
Rain Down On Me - RTZ
After the Rain - Nelson
Raining In My Heartland - Lynyrd Skynyrd
Who Let In the Rain? - Cyndi Lauper
Walk Between Raindrops - Donald Fagen
No More Cloudy Days - Eagles
Apologies if some of these have already been mentioned.
Chocolate Rain - Tay Zonday
Blind Melon - No Rain.
Come Rain or Come Shine - written by Harold Arlen and composed by Johnny Mercer. Recorded by many artists including Ray Charles.
Same Old Lang Syne by Dan Folgelberg. For most of the song it’s snowing, but at the end it turns into rain.
Edit: apparently it’s Same Old, not Same Auld
None of which I have ever heard of. To a kid in the suburbs in the 1970s, Rainmaker meant the Partridge Family.
It was as close as I ever came to a theme, I guess, although it was just kind of a coincidence that both of these seemed to be about rainy days when I wrote them.
The what?
Priscilla Ahn Rain. Bonus points: includes the lyrics to the children’s rhyme.