Despite being a big music city, I'm not sure that Austin has an official song. I mean, there are definitely movies that I think of that are the official Austin movie. Something like Slacker or Dazed and Confused, but I don't know that we have a song. If you go to Youtube and you search for songs about Austin, most of them are country songs and ones that I've never heard of before.
Oddly enough, this one here by Jimmy Fallon is pretty good, even if I wouldn't ever consider myself to be a country boy, I have to admit, this did make me smile when I saw him do his show here a couple of years ago.
• Video Link
As for something that mentions Austin, or rather, Travis County, I'd have to say Gary Clark Jr.'s "Travis County" is what I probably think of, not so much because of what he's talking about, but because Gary Clark Jr. is a hometown guitar hero. He was also playing in that Sweet Home Chicago video that SIGSEGV posted.
• Video Link