It’s Raining In My Heart - Dames At Sea (Broadway musical)
In the Early Morning Rain - Gordon Lightfoot
Angel Eyes - Sting
Why Does it Always Rain on Me - Travis
Kentucky Rain - Elvis Presley
“Wasn’t That a Mighty Storm” – American spiritual
“I Wish It Would Rain” – Wreckless Eric
“So Much Things to Say” – Bob Marley (When the rain fall,
it don’t fall on one man’s housetop…)
“Kaya” – Bob Marley (Got to have kaya now,
for the rain is falling… I feel so high, I even touch the sky,
above the falling rain….)
“MacArthur Park” – Jimmy Webb (Someone left the cake out in the rain…)
How could we all have forgotten “Singin’ in the Rain”?
There’s also “The Rain in Spain” from My Fair Lady, and “Soon It’s Gonna Rain” from The Fantasticks.
“Walking in the Rain” (Ronettes)
“Didn’t It Rain” (Mahalia Jackson)
“Feels Like Rain” (John Hiatt)
“Box of Rain” (Grateful Dead)
“Rainmaker” (Traffic)
“I Am the Mercury” by Jimmie Spheeris ends with a lovely coda of “Let it rain, let it rain, let it rain on the mountain”.
Oh—almost forgot “Rainy Day In June” by The Kinks.
If that doesn’t work for you because it’s currently October, not June, feel free to substitute Stevie Wonder’s “Summer Soft”
Always found this one moving:
Dancing in the Rain – Andrew McKnight
“Here Comes That Rainy Day Feeling Again” - The Fortunes
“Somewhere Over The Rainbow” - Israel Kamakawiwo’ole
Can’t have a rainbow without rain. 
REO Speedwagon – “Ridin’ the Storm Out”
Don’t suppose you’re interested in Slayer’s “Rain in Blood”
From 1960 - Dee Clark - “Raindrops” (oldies radio stations used to play it, before 80’s songs became the The New Oldies '…it must be raindrops…so many raindrops…feels like RAAINDROPS…falling from my eye-eyes…).
I like the cover art on the CD for ‘Jazz for a Rainy Afternoon’, and I may get it. Just because I like the idea of jazz for a rainy day.
Can’t believe I forgot Bob’s “Buckets of Rain”
Fleming & John - “Rain All Day”
Rain - The Cult
Rain - The Beatles
Into Each Life Some Rain Must Fall - The Inkspots
Spring Rain - The Go Betweens
Looking through the stack of 45’s in the back of my prehistoric cave…ah! Crying In The Rain, by the Everly Brothers!
And I seem to recall him doing some little ditty called “A Hard Rain’s A-Gonna Fall”…
… also, can it be that we haven’t yet mentioned “Stormy Weather”? Keeps rainin’ all the time…
Cracker - Another song about the rain
Songs: Ohia - Didn’t it Rain
“Laughter in the Rain” (Paul Anka)
“Light Rain Blues” – Taj Mahal
There are. If I was in an English program, I’d write a paper on rain as a motif in Rock and Roll/Pop Music. As compared to say, the sea as the primary symbol of nature in Romantic Poetry.
Couple more slight streches:
“Stormy Monday” by a whole lot of people
… and there’s a song about a guy who likes, among other things “being caught in the rain” (he’s also “not into champagne”). Not to spoil it, but turns out someone he knows also has similar likes and dislikes, including a particular rum and coconut mixed drink.
Raintown - Deacon Blue
Happy When It Rains - The Jesus and Mary Chain
I totally forgot:
Nick Cave and the Bad Seeds - Tupelo one bad ass rainstorm