Obviously [del]totally ripped off[/del] inspired by this thread
- Stormy
- Stormy Weather
- Good Night Irene (at least for the upcoming one)
Bad Moon Rising.
Like a Hurricane… Neil Young and Crazy Horse…
Rock You (like a Hurricane)
Windy
It’s Raining – Rick Derringer
First one I thought of.
How about “Trying To Reason With Hurricane Season” by Jimmy Buffett? Or for that matter, Jimmy’s “Apocalypso” and “Kick It In, Second Wind.”
So. Central Rain - R.E.M.
A Mighty Wind by the Folksmen, Mitch and Mickey, and the New Main Street Singers.
Hurricane - Bob Dylan. Although it’s not about a weather system.
Five Feet High and Rising - Johnny Cash
There’s a whole family of folk songs that have variations of “oh, the cruel rain and the wind” as their refrain. They’re mostly not actually about hurricanes though. Bob Dylan borrowed this for “Percy’s Song” (here performed by Fairport Convention).
A different song that also has that as a refrain is “Cruel Wind and Rain” by Jim Lauderdale.
And, of course, “Blowin’ in the Wind”.
And Frank Sinatra’s “Summer Wind”… also not about a storm but it’s a beautiful song.
Dylan also had “A Hard Rain’s A-Gonna Fall.”
Gimme Shelter - Stones
“Blow Away” - George Harrison
“Concerto for a Rainy Day” - ELO (consisting of “Standing in the Rain”, “Big Wheels”, “Summer and Lightning”, and “Mr. Blue Sky”)
Riders on the Storm - The Doors (OK, a bit of a stretch!)
Riding the Storm Out - REO Speedwagon (Technically about a snowstorm though)
Louisiana 1927 by Marcia Ball
I Love the Rainy Nights
Rainy Days and Mondays
“It Never Rains in Southern California” 
Mary Chapin Carpenter, Down at the Twist and shout.