How many hours have bands spent making this crucial decision?
I listen to a lot of oldtime stringband music. Favorite names I’ve run across include:
Ben Steele and his bear, Hans (mentioned in liner notes to a Bob Carlin album called “Where’d You Get That Hat?”)
Cutting Up Gumby (a perennial favorite local oldtime band in my neck of the woods)
Patty Hearst Shifter and Southern Culture on the Skids (two local rock bands)
Got any good names that never hit the mainstream?
Don’t know why, but “Road-apple” always seems to stand out on its own.
djm
“Cloister Folk”.
(No, I haven’t come across it, but I can dream.)
Oi Polloi comes to mind. Also Minor Threat and Mill a h-Uile Rud.
This topic is the bain of my life.
Some old indie band names (warning: some in poor taste - no offence intended. I didn’t invent them):
The Jesus and Mary Chain
Jegsy Dodd and the Sons of Harry Cross
Half Man, Half Biscuit
Sultans of Ping FC
Extreme Noise Terror
Scraping Fetus off the Wheel
Very Things
The The
Spontaneous Combustion - they have their mikes in old brass fire extinguishers (great band, too).
Rumpke Mountain Boys (Rumpke is our local garbage collectors and dump).
Hey, I’ve got one or two of their CDs! Love the name, and the trashy photo layouts. Priceless. 
For the non-Gaelic-savy, “Mill a h-Uile Rud” means “Destroy Everything.”
Some other names that have given me a chuckle over the years:
The Dead Sea Squirrels
John Cougar Concentration Camp
Gang of Four
Throbbing Gristle
Akira S. E As Garotas Que Erraram
Jackie-O Motherf**ker
I Can Lick Any Sonofabitch In The House
Los Amigos Invisibles
The Foggy Notion Boys
The Durutti Column
The Skoubhi Dubh Orchestra
The Stiff Kittens (an early name used by Joy Division)
The Happy Valley Dance Band (an early name used by New Order)
The Smiths
Honorable mention goes to Sleater-Kinney in the “best band named after a freeway exit in Olympia, Washington” category.
I always wanted to name a band Persistence of Vision.
I thought about Absence of Alice.
I was going to mention The Dead Sea Squirrels. Hmph. Anyway, how about Carnival Knowledge and The String Beings.
A couple band I actually like… always found the names amusing.
Neutral Milk Hotel
The Angry Samoans
There is a basic design for a shuttle called The Colonel Mustard Space Ferry. I thought that might make a good band name.
The Kosmic Drainpipe
Although we have been holding out for a major label deal, we may go indie… 
One of my favorite’s is a Country/heavy metal band called
Hayseed Dixie (say it a couple of times…)
Yup, they do covers of AC/DC songs in bluegrass style. Actually, they’re pretty good, and the songs make the transition rather well…
Goose Creek Symphony (local southern rock band that does a redneck concert up here every year by the lake.)
The 16yo and her buds are planning a band called “HELL NO!” Even have a bunch of shirt designs made up, but still looking for a lead. (Daughter wants to be the bass player.)
“Lunch at the Dump” (A real bluegrass band from New Hampshire, USA)
“Lugworm Bait”
“Loose Chippings”
I’m hanging out for:
The Sebaceous Cysters.
There was a metal band in the 90s called Pi**ing Razors which I thought was a pretty good name for a metal band, more original than most anyway.
There’s some grind and death bands that have some pretty offensive names which I guess I should say are neither big nor clever. Anal C**t is probably the most offensive one just for use of the C word alone.
A friend of mine has started a band called Cobra Kai which makes me insanely jealous (been trying to think of a good name for my band for ages).