A line he stole (sorta) from The Meatmen. LONG LIVE TESCO VEE!
ETA: hey yeah, The Meatmen count here, many times over actually: Meatman, Meatmen Stomp, TSOL Are Sissies, One Down Three To Go, ABBA God & Me, Rock ‘N’ Roll Juggernaught, etc., etc.
Captain Beefheart & His Magic Band: “The Blimp”
It’s the thing that’s gonna make Captain Beefheart and His Magic Band fat, Frank!
The Mothers of Invention: “Motherly Love”
The Mothers got love that will drive you mad.
Leonard Cohen: “Famous Blue Raincoat”
Sincerely, L. Cohen
The Who had a song that asked “Who are you?” alot.
Liz Phair, Rock Me
“Your record collection don’t exist
You don’t even know who Liz Phair is”
Devo mentions their name in a number of songs, but usually it’s used as an adjective (i.e. “we’re all Devo”), so that prolly doesn’t count, but I thought of it so I’m posting it anyway. 
The Clash did Radio Clash
Pink Floyd, Have a Cigar
“By the way, which one’s Pink?”
They Might Be Giants had a song called They Might Be Giants.
Skid Row yelled their name in, i want to say Youth Gone Wild?
Janet Jackson in Nasty Boys
“No, my first name ain’t Baby. It’s Janet, Miss Jackson if you’re nasty.”
Veruca Salt, sort of, in Volcano Girls, referencing the guitarist and a previous single
“Here’s another clue if you please, the seether’s Louise.”
Which of course is a reference to the Beatles doing in Glass Onion.
Tabloid footprints in your hair
Tabloid footprints everywhere
We can’t be silent
'Cause they might be giants
And what are we going to do unless they are?
Does that count as a name check? I thought they were making fun of greedy record executives who wanted to sign the band but didn’t know the first thing about them.
This is Radio Clash from pirate satellite
Does Sgt. Pepper’s Lonely Hearts Club Band count?
The Meatmen RULE!
The Jon Spencer Blues Explosion does this often, too. Particularly on Orange, I can remember:
“Sweat”- “That’s the sweat - of the Blues Explosion”
“Blues X Man”
“Flavor”
The Monks had “Monk Chant”, though it only name checks them in the title.
And the best one, in my mind is from The Dicks’ classic song “The Dicks Hate the Police”:
People tell policemen
they’ve met their match.
They placed on desert sand
the Dicks were hatched.
In the Beatles song “Glass Onion,” in addition to referencing lots of their songs, John Lennon sings “the walrus was Paul”. Also, this might not count, but in their song “Come Together,” each stanza describes a different band member - but they use no names, just “he” for all of them.
ETA: Also, in the Simon and Garfunkel song “Frank Lloyd Wright,” Frank Lloyd Wright is a metaphor for Garfunkel. But again, it might not be specific enough.
The Yellow Balloon had a hit single with “Yellow Balloon.” The B side was “Noollab Wolley”
Alice Cooper had the lyric “She asked why the singer’s name was Alice” in “Be my Lover.”
“The Intro and the Outro” by the Bonzo Dog Band name checks all the members of the band – plus a few others. 
Shakira in “Hips Don’t Lie.”
Lady Gaga and Red One in “Just Dance.”
Make my funk the P.Funk, I want my funk uncut.
You are obviously a person of towering intellect and impeccable taste!
I am pleased to welcome you into my permanent circle of friends: Gabba gabba we accept you, we accept you one of us!
Back in the 80’s **Wang Chung **proposed that Everybody Have Fun Tonight but really what you want is that Everybody Wang Chung Tonight. We did. It wasn’t so bad.
Also a deep cut from **Steely Dan’s **Countdown to Ecstasy, “Show Biz Kids” has the line: “They got the shapely bods. They got the Steely Dan t-shirt.”
The Hold Steady…about a hundred times on their first record.
“Oh Well, wherever, wherever you are,
Iron Maiden’s gonna get you, no matter how far.”
“Watching them crumble, watching them crawl
Iced Earth has risen - nature will fall”
“In a place of fire eternal, left in flames.”
“Gonna keep on burnin’, always will
Other bands play, **MANOWAR **kills!”
“Everybody, everywhere in the whole world
Gamma Ray, Gamma Ray, Gamma, Gamma, Gamma Ray
Made of your existence Gamma Ray money and our failing
I’ve waited for - to find Gamma Ray.”
“**Annihilator **(this machine) Annihilator
**Annihilator **(this machine) Annihilator
**Annihilator **(now you’re dead) Annihilator
**Annihilator **(now you’re dead) Annihilator”
“I been mothered, fathered, aunt and uncled
Been Roy Haleed and Art Garfunkeled”
The early '70s Australian band Daddy Cool (better known for “Eagle Rock”) had a hit with “Daddy Cool” (not to be confused with the 1976 Boney M song of the same name).
I seem to remember Roxette calling out their name on a song whose name escapes me for the moment. Not even sure if it was a single or an album track.