Not a popular theme. The only one I can think of is “It’s a Bore!” from Gigi.
So Bored - Wavves
Plume - Smashing Pumpkins
Bored to Death Theme Song - Coconut Records
No Rain - Blind Melon may qualify, or you could say it’s all about drugs. Eh, often they go hand-in-hand.
Bored - The Deftones
Kim, You Bore Me to Death - Grandaddy
I’m tempted to write “The Pixies” as a whole.
Song 4 You - Ben Lee. This one’s not so explicit, but the chorus is “blah blah blah blah, blah blah blah blah blah”
Iggy Pop: I’m Bored
X-Ray Spex: I’m A Cliche
“Bleach Boys” by The Dead Milkmen.“I’m so bored I’m drinking’ bleach”
The Clash: I’m So Bored With The USA
The Adverts: Bored Teenagers
Not sure if this counts. Bruno Mars The Lazy Song
I’d venture to say you won’t like this, but I’m on holidays, and I’m bored: (nsfw, I spose)
when I’m bored
Fluttershy version:
“Alone Again, Naturally” by Gilbert O’Sullivan. Maybe it’s more about depression, but I’m not sure where the line is between the two.
“I’m So Tired” by the Beatles.
“Counting Flowers on the Wall”, Statler Brothers
What did Madeline Kahn sing in Blazing Saddles?
I’m tired, tired of playing the game…
Sparks " Nothing To Do"
Could We Be So Bored by Weird Paul
Bored With Beauty by Facepuller
It’s Easy To Get Bored by Helmet
Boredwitcha by Powerman 5000
I’m So Bored With The U.S.A. by The Clash
Boredom by Buzzcocks
Dirt by The Stooges
I’m Bored by Iggy Pop
Bored by Destroy All Monsters
Glass by Gang Of Four
I might have some more I remember later. If I get bored, I’ll think about it some. 
ETA: Ddamn, it took me a few minutes to make that Gang Of Four vid, and a few people beat me to a couple of these tunes!
Nearly forgot The Boredoms themselves, notably “Boredom with God on Noise (Borestafari!)”, “Bore Now Bore”, “Hey Bore Hey” and “Greatborefull Dead”: here’s a sample of Osaka’s finest…
Boredoms rule! Might as well mention Bored Elephant by Melt Banana (sorry, I can’t find a video and I’m too tired to make one tonight).
And it reminds me that I forgot to list Attention Please by Boris.
“Tired of Waiting for You” by The Kinks.
“Battle of Who Could Care Less” by Ben Folds Five.
The Kinks had a few boredom songs:
Predictable (lyrics)
Repetition (lyrics)
Cliches of the World (live version; lyrics)
Nine to Five (lyrics)