No Weird Al? But he has White & Nerdy!
Two recommendations:
The Galaxy Song by Monty Python
All songs by MC Hawking
I wrote a really nerdy song when I was studying quantum mechanics in school . . . sometime I’ll have to record myself singing it and stick it on youtube, so you all can get a good laugh at my expense.
Oh, what the hell. Here are the lyrics:
Groundstate
I’ve tried to make you happy, but it seems a hopeless mission
when all your conflicting wishes are in a superposition
and all that you demand, you can reverse with no compunction
as your mood can shift as quickly as the Heaviside step function
Knowing what you want from me is impossible, and then some
I’ll sooner know exactly both position and momentum
The discrepancies between our needs could simply not be greater
They’re discrete as the eigenenergies of the harmonic oscillator
Oh I’ve tried and failed to figure out a way to make it better
I’ve done my best to follow all of your laws to the letter
Yet it seems eventually we always end up fighting
I guess anger is your groundstate, and you’re simply deexciting
I hate how you insult me and you act so inconsiderate
You weren’t thinking of my symmetries when you called me a degenerate
But if I say one word back to you, you’ll just say “See you later”
You think I’m as easy to exchange as two commuting operators
When I say “Let’s discuss this” why do you hear Open Season
I’m still searching for a basis in which you’re subject to reason
Why do I even bother? Each attempt makes me feel dumber
When under every perturbation you have no good quantum numbers
Oh I’ve tried and failed to figure out a way to make it better
I’ve done my best to follow all of your laws to the letter
Yet it seems eventually we always end up fighting
I guess anger is your groundstate, and you’re simply deexciting
I guess anger is your groundstate, and you’re simply deexciting . . . .
Lute Skywatcher: Weird Al’s fine (and is likely to have one on the comp), just I didn’t want loads of responses suggesting the first artist that came to mind instead of going for something more unusual.
runner pat: Nice one.
I haven’t got to listening to some of the other suggestions yet.
I don’t like rap, so a rap song would have to be seriously exceptional for me to put it on.
I forgot completely about Trout Fishing in America. They’re a folk duo who have a number of wonderfully geeky songs.
Dead Egyptian Blues
Two Brains
The Day the Bass Players Took Over the World
I’d also suggest that songs that tell history are geeky, too. So Boney M’s Rasputin counts. (As does the cover by Boiled in Lead.)
I can’t stand (most) rap, but I still like nerdcore because it’s generally smart and funny and cleverly worded (not to mention geeky). (Please, no offense intended to “real” rap/hip-hop fans–just not my cup of tea for the most part.)
Give MC Lars a try before you reject him–he’s pretty good. 
Oh, and The Gothsicles are good too–industrial/EBM-type music based on old video games and other nerdy topics. How geeky is that?
How about “New Language” by The Pursuit of Happiness? Any song which uses words like “benign” and “efficacious” has to be nerdy, right?
I’m at work and can’t access any video or audio sites (blocked by proxy), but here are some you should be able to google.
The Picard Song (lots of Jean-Luc Picard quotes set to techno music)
The Device Has Been Modified (lots of GLaDOS (from Portal) quotes set to techno music)
The USS Make Shit Up (Star Trek parody song by Voltaire)
ETA - Just thought of 2 more! There’s also Star Trekkin’ (another Star Trek parody song) and a song by Bree Sharp that I don’t remember the title to, but the lyrics go “David Duchovny/why won’t you love me/I’M GONNA KILL SCULLY!” It’s about a stalker-fan obsessed with David Duchovny from X-Files.
Great geeky rap? Try SNL’s “Lazy Sunday”.
More for sound than content: anything by Jonathan Richman and the Modern Lovers.
Or “Ariel” by Dean Friedman.
“Star Trekkin’” by The Firm.
“The Ballad of Bilbo Baggins” by Leonard Nimoy.
Conventional Lover by Speck is a catchy little song.
I don’t mean to boast
I don’t mean to brag
but I’m a man whose only issues come in mylar bags
Hell, if you’re going to use a Portal reference, just get “Still Alive.”
“Aperture Science: We we what we must, because we can.
For the good of all of us, except the ones who are dead.”
“And the science gets done, and you make a neat gun,
For the people who are still alive.”
The first song that lept to mind when I read the thread title is the Pet Shop Boys’ “Opportunities (Let’s Make Lots of Money)”, although the narrator seems to be more a conman than a real genius/geek. This song is used as the theme to the reality show Beauty and the Geek, though.
I agree with the above suggestions of Jonathan Coulton and Flight of the Conchords. Another Jonathan, Johnathan Richman, may also have some good tunes for your compilation – I recommend “Vampire Girl”.
The Songs To Wear Pants To site is a great source of geeky music. The site’s creator is a musician and composes short songs at the request of site visitors. They cover a lot of different subject matter, but a lot of them are pretty geeky! Many songs are free, others are 99c downloads.
I’ll nominate '39 by Queen. I’m pretty sure it’s a song about time dilation and that’s about as geeky as you can get.
I have to (third, fourth, whatever) the MC Frontalot recommendation. I think even people who “don’t like rap” would like him if they’re nerdy. Check out his songs page where you can listen to and download for free some of his songs. I recommend “It Is Pitch Dark” (a paean to old-school text adventure games), “Crime Spree” (a parody of “gangsta” rap, in which he brags about how tough he is because he tears tags off mattresses, etc.), “Message No. 419” (about Nigerian E-mail scam), “Pr0n Song” for a NSFW description of porn addicts of the Internet age, and if you want a fun twist, he retells the Little Red Riding Hood story from three perspectives in “Start Over” - very kid friendly, too.
This one might be even better.
It was actually just called “David Duchovny” and rumor has it that the real David Duchovny was spotted at a red light singing along to it after the song came out.
Laurie Anderson has some nerdy/geeky stuff. I, a licensed nerd, like her stuff a lot. It’s … different … in a good way.
How about a series of fish puns?
“The Book I Read,” “Don’t Worry About the Government,” and other early Talking Heads material.
The Juno soundtrack.
The group - Three Dead Trolls In A Baggie
Behind The Scenes At Microsoft
Mom’s Coming to My House
Every OS sucks
They have more Geeky songs.