I recently moved after living around the corner from the Music Box for 8 years, so I've been to most of the restaurants in the area many times. I'd recommend the following, all of which are within a half mile or so of the theater:
Tango Sur - For my money, this is the best value for steak in the city. I'm partial to the vacio cut (flap meat). Just make sure they cook it to your liking, the Argentines like it rare and the kitchen can be inconsistent when it's busy. The servers never have a problem correcting a mistake.
Coalfire - one of my favorite pizza places in the city. The pies can come out really quick, which is nice if you're pressed for time before the show.
Toons - Really good bar food. I especially like the Cuban sandwich and their wings. The Cuban isn't a 100% traditional version (it includes pulled pork) but it's better than almost anywhere else in Chicago.
The Roost - Good Nashville style hot chicken (non-spicy fried chicken also available) and biscuits.
Tac Quick - Great Thai, a little further from the theater than the others but still walkable. Note newish location on Irving (used to be around the corner on Sheridan). I love the crispy on choy.
Southport Grocery - a breakfast and lunch spot that's worth the often long waits. Sadly only an option if you're going to a matinee.
A couple places I wouldn't recommend:
Crosby's Kitchen is pretty good for a neighborhood place and is really popular for families with small children, but I wouldn't make a special trip there.
I've only eaten at Tuco & Blondie once. It was just OK, I don't plan to return.