Good one. I always thought Robert Plant was influenced by her.
Geddy Lee
Bonnie Tyler
Joe Cocker
Dolly Parton
Tom Waits was the first to come to mind; already mentioned. Same with Liz Frazer.
But how about Björk?
Good pick. The voice of Popeye taken to amazing levels of virtuosity.
Billy Corgan, too. A thin, nasally voice, instantly recognizable. I couldn’t stand the Pumpkins for several years because of his voice, then I somehow flipped to just loving them, and enjoying his voice in context.
Peabo Bryson’s voice always sounded weird to me.
Heh, yeah. The first time I ever heard the Decemberists I was like “WTF kind of accent is that?” and went googling to see where he was from. NOT the answer I was expecting.
Natalie Merchant has a weird voice. Not exactly flat, but something has always just sounded… off… about it to me.
Keith Moon (you didn’t say it had to be good 
)
Robert Wyatt – reedy and thin, and his lyrics never rhyme, but it’s distinctive, and works nicely.
Mark E. Smith, of course!
I would say VU’s Nico qualifies. It is probably more how she sings as opposed to a particular timbre in her voice.
Kim Gordon of Sonic Youth. Not sure she is really singing but I like it
Les Claypool of Primus
Joe Walsh. I read one reviewer describe him as sounding like he’s singing somewhere across the street.
Macy Gray. She can make a suicide note sound sexy!
Tyler Glenn of Neon Trees has a very noticeable speech impediment. It sounds like a mild Rhotacism. Not a usual trait for a lead singer.
No one really sounds like Bono.
No one really sounded like Prince, either.
Eric Woolfson, the singer of most famous Alan Parsons Project songs, sounds like no one else, really.
Antony (now “Anohni,” I think) or Antony & the Johnsons.
The Swedish singer Miss Li. I first heard her song “I Can’t Get You Off My Mind” in a commercial and was intrigued. Her voice is very distinctive, I think. I really like this song.
Macy Gray and Fantasia both have distinctive voices in the same squeaky way. Leon Redbone was one of a kind.
Charlotte Church certainly had an unusual voice for a 14 year old.
This is a terrible video of Charlotte Church but the guy states her age right at the beginning, so I used it.
Mike Heron and Robin Williamson from the Incredible String Band.