I’m a big Me First fan (love their “Hava Negila”), but I think my favorite might be Catch 22’s “American Pie.” It sounds like a drunken ska band just repeating the chorus over and over, because they can’t remember the words. And I’m willing to bet that’s exactly how it went. 
Ooops, I forgot to mention a fine weird cover band, Dread Zeppelin. Half-Elvis, half-reggae version of Zeppelin and other classic rock songs - who could resist?
Richard Thompson covering “Oops, I Did It Again” on 1,000 Years of Popular Music, and Hindu Love Gods (Warren Zevon + R.E.M. - Michael Stipe) doing Raspberry Beret on their eponymous disc.
The Flaming Lips did a great cover of Kylie Minogue’s I Just Can’t Get You Out of My Head that slows it way down and makes it all dirge-like and sad. Much better than the original.
Some covers are better than the original.
Moon River by Andy Williams
You Showed me by the Turtles
Never My Love by Fifth Dimension
As Tears Go By by Marianne Faithfull
All I Really Want To Do by Cher
Blinded By The Light by Manfred Mann’s Earth Band
Chains by the Beatles
Feelin’ Alright by Joe Cocker
Hush by Deep Purple
In the Midnight Hour by the Rascals
Love Hurts by Nazareth
Love Potion #9 by the Searchers
Me And Bobby McGee by Janis Joplin
Mr. Tambourine Man by the Byrds
Proud Mary by Tina Turner
Red Red Wine by UB40
Respect by Aretha
You’ve Got A Friend by James Taylor
OK, I was with you until you hit this one. As far as I can remember, the original is Gram Parsons and Emmylou Harris. On what planet is that worse than Nazareth, for God’s sake? 
Oh! How could I forget?! Mitch Miller, of “Sing Along With Mitch” fame, did a cover of John Lennon’s Give Peace a Chance. I’ve actually heard it, once, about ten years ago.
The horror! The horror!
By the way, LindyHopper, Roy Orbison wrote and made the first (and IMO best) recording of “Love Hurts.”
Thanks, but most of these wouldn’t be considered “strange”.
I’m listening to another good example, that “Lean on Me” dance/reggae cover that came out in the '80s.
Shang Shang Typhoon did…something to Let It Be.
Reeeeally? Learn something new every day, I guess. I’ve never been much of a Roy Orbison fan, myself. Just don’t really like his voice. So I’d have to go with the Parsons/Harris version. IMHO.
Ooh! I’ve got another one! Frente’s cover of “Bizarre Love Triangle”! New Order enthusiasts hate it, but I actually like it better than the original, and it’s way different.
JerH beat me to Dread Zeppelin
Richard Cheese is one of my all time favorite cover bands, especially Closer orginally done by NIN (explicit lyrics).
Me First & The Gimme Gimmes covers aren’t strange, really, but IMO they’re almost always better than the original. In particular, “Danny’s Song,” “Rocket Man,” and “Fire and Rain” I can listen to now and think, “you know, those were actually really good songs if they’d just been done the right way.”
As for the OP: The Cardigans still win, with their Black Sabbath covers. “Sabbath Bloody Sabbath” is just a neat little jingle, but “Iron Man” crosses the line over to brilliance.
Strange covers:
Klaus Nomi doing “You Don’t Own Me.” It’s either traumatising or hilarous, depending on what substances you’ve ingested.
I don’t know if this counts but I was watching Malcolm in the middle where the Dad falls into the dance dance revolution craze and my jaw about hit the floor when I heard a techno-y version of Dan Hill’s Sometimes When We Touch. No clue who’s responsible for it, but it was oddly disconcerting.
In the UK on BBC Radio 1… Jo Whiley has Live Lounge sessions, with bands of various musical styles…
everytime she does this she asks the band to play a cover version of a song that differs from their usual style ( at least thats how I understand it ) this has thrown up some real gems…
like
Embrace - doing D12’s - “How Come” ( They only do Eminem’s verse - but when you think about it, it’s the only one that works as a song )
Will Young & Razorlight both did different but both good versions of “Hey Ya”
Jamie Cullum did a GREAT version of “Frontin’” which made me appreciate the original more - - but still prefer the Cullum version…
ummm… loads of others… I’m sure somewhere on http://www.bbc.co.uk/radio1 you can find a listing or something.
apart from that, on the Anthony Stewart Head album. “Music For Elevators” - tehre is a very strange cover of “We Can Work It Out” by the Beatles…
Plus… no-one’s mentioned “Mad World” - from Donnie Darko yet
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Johnny Cash’s cover of the Nine Inch Nails song “Hurt” is a strange cover but it definitely comes out as a good song. I guess it helps that I like NIN already and like hearing a different take on songs by artists I like.
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I was surprised when I heard a Punjabi Mc song (Mundian To Bach Ke) and then Jay-Z started rapping over the top of it. I was listening to an independent radio station at the time and thought it was a live DJ mashing the two together but it was an authorised mix and top-40.
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Dsico (A Sydney DJ) does some very odd remixes of popular artists such as Nelly & Missy Elliot.
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I also have to agree with the Tori Amos cover of Eminem’s “Bonnie and Clyde '97” as being strange.
I’m not really sure if the Punjabi MC and Dsico remixes technically count as covers as they use material from the original song .
The all-time king of weird cover versions has to be Rolf Harris; I actually liked his skiffle rendition of Bohemian Rhapsody and his cover of Stairway to Heaven was good if for no other reason than it was fun to watch diehard Led Zep fans working themselves up into a froth of offense.
It’s strange and dark, but I really like A Perfect Circle’s version of John Lennon’s “Imagine.” Similarly, Type O Negative doing Seal and Croft’s “Summer Breeze”. Strange and fun would be Travis’ cover of Brittney Spears’ “Baby One More Time” and Julianna Hatfield’s cover of Weezer’s “Say it Ain’t So.”
Aha! What most people have never heard (although it’s widely bootlegged) is that Page and Plant returned the favour and did Rolf Harris’s “Sunarise”. It’s pretty good. Strange but good.
They did it on that Aussie tv show where everyone had to cover “strairway to heaven”.