If you walk South of Wrigley on Clark Ave, there are many, many local mom-and-pop restaurants. Plenty of them are pretty good. Any particular cuisine? Beer or just food? Here are a few suggestions:

Goose Island Restaurant serves locally brewed namesake beer. Though certainly not unknown to tourists, it's pretty good and is frequented by locals: http://bit.ly/Mqxh3T

I frequently stop by El Jardin Mexican Cafe at 3401 Clark Ave before a game - it has pretty good food and nice courtyard seating.

You can try Byron's Hot Dog Haus just North of Wrigley Field (1017 West Irving Park Road). Their claim to fame is that their dogs were served at the White House...though I personally think they're overrated.

Another good beer place is Uberstein (http://www.ubersteinchicago.com/), though it is neither mom-and-pop nor old.