Need a Song I can Cover

I'm not the biggest fan of covers because the original artist usually did it better than it will be, however I know that theory is full of holes. Anyway, not to be a coward, I am trying to acquiesce to wishes from the group and decide on a song I could do. It has to be something that works pretty much acoustically, so most Led Zeppelin is out, and I don't think Doors will fit the general tone. I hear old songs and think about it sometimes but just haven't settled. I'm sure even closest associates doubt my ability to pull it off, but I know better. The right thing and it will be legendary or, at worst, mediocre.
I'm toying with Catch the Wind, but I think maybe something else.

Something sexy and insane, yet sedate.
It must have an identifiable melody, not like the stuff Merv Griffin used to sing. Did anyone else ever see his show years ago? He would sing stuff that had no melody or anything. It made me worry about the composer. Holy smoke. I couldn't imagine anyone actually being excited about writing whatever that song was. Was it a song or maybe Merv was given to fits of insanity which led him to sing non songs. He seemed like a nice guy. Some of the tunes were, I don't even know the word.
Not tappers, those.
Help me, but no Merv Show debacles.
Do any of the thousands of readers of my blog have an idea?