Austin has a motto - "Keep Austin Weird". You can see this printed on T-Shirts, rear bumpers of the cars, at restaurants, almost everywhere. You can get this feeling in Southern part of the city, just go for a window shopping on South Congress Avenue.
Or just compare 2 Chuy's restaurants - one from the North and one from the South and guess where the food will taste better...
Vince Hannemann. For me, he and his cathedral is one of the best examples for the weirdness of Austin.
He got the idea from a trip to Mexico and he built his own cathedral from junk, trash, unused parts that he found and got from other people. In 2010, he was complained and code enforcement came, and they told, he have to have a building permit.
In order to get the sign off from an engineer and the building permit from the city, he rebuilt the cathedral (the first part of it has gone).
Now, it is open for visitors. I'd definitely recommend to go there. You will get a real picture from Austin's Weirdness.