Oh, and I almost forgot: The Sultans of Bing
Aww, the first one I thought of is in the OP - Brian Jonestown Massacre.
Cloud ten
Pist*on
Modern Day Zero
Velvet Teen
Gemma Hayes (I think it’s a play on Gamma Rays, anyway)
The Plimsouls
'splain?
A plimsole (also spelled plimsoll or plimsol) is a sneaker. The “sole” syllable of the (variant spelling of the) word is itself a pun – the word ultimately derives from the name of Samuel Plimsoll, a merchant memorialized in the nautical term Plimsoll mark.
I can’t believe I didn’t think of this one sooner: Flash and the Pan
One more that has yet to be mentioned: Terrorvision.
Fitz and the Tantrums
This is an eleven year old thread.
The Zombies? 
We’ve got Spam, Spam, Spam, Spaghetti and Spam.
Reported.
Stephin Merritt’s side project, “The Gothic Archies.”
(Also had a project called “The 6ths,” named to aggravate lispers.)
Also, the solo act known as “Kurt Vile.”
It wasn’t formed in 2006: 2NE1, a K-Pop girl group, can be read as “twenty one” or “to anyone”.
[Moderating]
Although this thread was bumped by a spammer, there have been three legitimate on-topic posts after that, so it looks like there’s interest in continuing the discussion. So I’ll keep it open.
Great. Disappearing the spammer’s post makes it look like I bumped it.
Did I miss a mention of The Beatles?
Which is why I make the reporting post like that.
Elvis Depressedly
Peace Corpse
The all-female Iron Maiden tribute band is The Iron Maidens
Not a band name, but original Beatles drummer Pete Best released an album called Best of the Beatles.
When I first heard the name, I thought The Allman Brothers Band was a pun: “All men (are) brothers.”