Here are some good spots close to the convention center
Sandy's Hamburgers on Barton Springs right across from the Convention Center has cheap, good old fashioned style burgers. Thursday night is their $5 combo deal (burgers, fries, and a drink for 5 bucks). Their frozen custard and hot fudge sundaes are great.
La Mexicana Bakery is on South 1st and open 24 hours a day. Pop in for some migas tacos and chips and salsa when you get the munchies late hours. Their tres leches cake and pastry cone filled with vanilla cream are my favorites also.
Polvo's Mexican is on South 1st next to La Mexicana bakery. It gets busy but is good also.
Black's BBQ is on on Barton Springs right across from the Convention Center also. Their brisket and pork ribs are good.
If there are any specific types of food or beer/cocktails you are looking for I can give you more recommendations.