We fly coast-to-coast at least twice a year and I like to pack enough for at least one full meal for DD plus snacks. I have to admit that I am a pack mule, so I probably over-pack, but I'd rather not have a hungry DD.
I usually pack cooked plain pasta (great finger food), cucumber slices, carrot sticks, little cubes of ham, hard boiled eggs, apples, strawberries and some of the other things PPs have mentioned. For snacks, DD loves fruit chewie snacks, freeze dried fruit, japanese rice cracker mix and Pringles (the individual serving sized and they don't get crushed!). I usually sneak in a dum dum lollipop too.
Not sure how old your kidlets are, but I also pack 2 juice boxes (one frozen and one refrigerated) to help keep everything cool and have never had a problem going through security.
Happy eating!