Stray Cats- “Stray Cat Strut”
Boston- “Rock & Roll Band” (Well, we were just another band out of Boston…)
Not exactly. The song “Motorhead” actually came before the band Motorhead. The song was written and recorded while Lemmy was still in Hawkwind; after he got booted, he wanted to name his new band “Bastard” or something like that. He was eventually persuaded to choose something more publicity-friendly, and settled on the name of the song.
Incidentally, the Motorhead version of “Motorhead” is waaay better than the Hawkwind version. I still wish that Lemmy and crew would do a version of “Silver Machine”–if they have, I haven’t run across it.
There was also this thread about songs in which the band, rather than reference themselves, proudly declares themselves to be some other band.
The Butthole Surfers, rather than write a signature song about the band, took the name of the band from one of their own songs. One of their early years schticks was giving the band a new name at every show (Dave Clark Five, The Dick Gas Five, Nine Foot Worm Makes Own Food, Vodka Family Winstons, Abe Lincoln’s Bush, etc.) One of their songs, Butthole Surfers, was a particular hit with the crowds and audiences would chant for it. Eventually they when it came time to settle on a name they went with “the Butthole Surfers,” since everyone was calling them that anyway.
Oh yeah…Lard has “Lard”.
“Talk Talk” by Talk Talk
“Move on” by No Doubt on their album entitled “No Doubt.” Back when No Doubt was ska/punk, not bubblegum pop.
“You’re Tripping” by The Queers features the refrain
“bring on the beers/
'cause we are The Queers”
“The Bitch Is Back” as sung by Tina Turner? 
(Naaaw, I luvs me some Tina!)
Not sure if this counts, but the Rolling Stones have covered Dylan’s “Like a Rolling Stone.”
The Clash have “Clash City Rockers” and “This is Radio Clash”
Then there’s Bob Dylan’s own “It’s Bob Dylan Time (AKA Time to Party Up in this Bitch)” 
“Van Dyke Parks” by Van Dyke Parks in his Song Cycle
“The Bride Stripped Bare by ‘the Bachelors’” and “The Intro and the Outro” by the Bonzo Dog Band.
The album “Quadrophrenia” by the Who has several songs that reference the band and its members.
Motley Crue managed to stick the names of Nikki Sixx, Tommy Lee and Mick Mars in their remake of “Smoking in the Boy’s Room” and are associated more with it than Brownsville Station. That’s no easy feat. 
“It’s Raining Men” by the Weather Girls.
I think they existed for the sole purpose of performing that song.
Hash Pipe
or
Buddy Holly
- Weezer
They took their name from the Muddy Waters song Rolling Stone… did they ever play that?
October Country…“October Country” 1968
Ace in the Hole by George Strait and the Ace in the Hole band.
And whose signature song would that be?
The theme to Rat Race performed by the Baha Men has their name spelled out in the song.
The Grateful Dead’s “The Music Never Stopped” is more or less about themselves and the whole Dead scene. Always got the crowd groovin’.