There's also an Italian grocery on Cross bay that's really good for a to go lunch. I can't remember the name, but it's on the west side of Cross Bay right next to an ice cream stand. The deli has sandwiches, but the prepared food is where it's at.
I like Vetro which might be more upscale than you are looking for. Excellent food and they have lunch specials which are pretty affordable. I am not sure if they are child-friendly but you might want to give them a call.
New Park has great pizza but pizza only. LaVilla is always a good choice pizza and a full menu. Aldo Restaraunt,Lennys, Sugarbun Bakery has a small lunch menu, In My opinion La Villa is the best choice with kids and for variety.