I'd say Under The Bridge for RHCP.
Aerosmith... Walk This Way.
John Mellencamp. Pink Houses.
Tom Petty... hmm. Running Down a Dream.
Van Halen. Gotta be Jump.
I don't care about Green Day, and neither should you.
I'm tempted to say November Rain for Guns and Roses, but I'll agree with Spiff on this one.
Pearl Jam... even though it isn't their best song, it's gotta be Jeremy.
ZZ Top... she's got Legs, she knows how to use them.

Anyway, for some others...

Deep Purple - Smoke on the Water
Boston - More Than a Feeling
Soundgarden - Black Hole Sun
Collective Soul - Shine
Journey - Don't Stop Believin'
Blue Oyster Cult - (Don't Fear) the Reaper
Foreigner - Cold as Ice
Loverboy - Working for the Weekend
Bad Company - Bad Company
Ted Nugent - Cat Scratch Fever
Queensrÿche - Silent Lucidity
Styx - Renegade
Yes - Roundabout
Supertramp - Take the Long Way Home