The Berghoff comes to mind immediately - maybe because of the other recent thread!

The Italian Village is not budget, but fits your other criteria and is an excellent place to eat.

If it's breakfast you want, try the place where I once saw Jimmy Carter 10 feet away from me eating with the dining room full of Secret Service (let me tell you, I was glad I didn't sneeze or blink!) - Lou Mitchell's on West Jackson Blvd.

The problem with downtown eating is that there really aren't many landmark places left. Most of them have been gutted and rehabbed, or simply torn down and replaced with sports bars.

The Pump Room may be an option, but that's Near North Side/Gold Coast.