Solo dining near the Indianapolis Hilton?
If you're looking for lunch, City Market at Market and Delaware (3 blocks east of the Circle) is basically a huge food court. They do close by 3 or so, and I'm not sure whether they're open on weekends or not. Ameer was always one of my favorites, and has been around for years and years. Another is Grecian Garden, but I have no idea whether they're still there or not. They're both cheap.
If you're willing to hike or take a cab/bus, the Rathskeller (German), MacNivens (Scottish) or Yats! (noveau Cajun/Creole) on Mass Ave would be good non-chain bets. I think it's about 2 miles to those.
There are two places I really like in downtown Indy: the Oceanaire and 14 West, both of which are located on the same block as the Circle Center, but with entrances from the street. I've dined at both solo (although, offered the choice, I've preferred tables rather than the bar at each, but that's up to you). One, the Oceanaire, specializes in seafood, which you don't want, so you should go to the other one, 14 West. You can read my report on 14 West in the topic at www.chowhound.com/topics/348586
14 W. Maryland St.
Indianapolis, IN 46204
(From the Hilton, walk two blocks south to Maryland, turn left, and it's on that block right after you walk under the walkway.)
30 South Meridian Street
Indianapolis IN 46204
(Just around the corner from 14 West.)