Basically anything found within one of the four major food groups is probably going to be a healthy option. Whole foods that have less processing generally will be healthier as processed items such as cheeses, pre-made frozen snacks or junk foods often are higher in fat and sodium and sugar and usually contain higher levels of preservatives and/or other chemical additives.
Think about things generally found around the periphery of your supermarket or foods that your great-grandparents would have eaten (i.e. fruits and veggies, meat 1 or meat substitutes 2 , eggs, cheese or yogurt, whole grain breads, bagels and crackers).
- Diced fruit and vegetables
- Whole wheat crackers
- Whole wheat toast or bagel with cheese or peanut butter
- Boiled eggs
- Low fat yogurt 3
- Pita and humus
- Homemade pizza snacks (sauce, toppings and cheese on top of a pita or crackers toasted in the oven)
The American Association of Pediatrics as a great info page 
1 Some processed meats like hotdogs or coldcuts can be very high in fats, sodium and preservatives so check the labels if you are concerned.
2 Tofu, veggie dogs and healthier options like turkey bacon can offer your kids the protein they need without the additional sodium and saturated fats of processed meat.
3 Younger children and toddlers are okay with reasonable amounts of higher-fat content dairy products as their bodies need the additional calories for growth and development but older children and adolescents can gradually switch over to lower-fat products.