Any bag will work, but we prefer backpacks because they make the best luggage for kids' carry-ons. I am a huge fan of Pottery Barn Kids backpacks when it comes to buying a toddler backpack and kids' carry-ons.
An abundance of snacks saves the day. This is especially true for toddlers! Some good snacks are granola bars, pretzels, goldfish, cereal, crackers, fruit squeezers, and fresh or dried fruits and vegetables. I make sure my kids each have their own water bottle. Fill it after you go through airport security.
You never know when your kids might need a change of clothes. For this reason, I always put a complete set of clothes (top, bottom, undergarments, and socks) in a plastic freezer bag in my kids’ backpacks. The extra-large IKEA freezer bags can fit older children's clothes and swimming items.
Be sure to pack headphones for personal electronics or for in-flight entertainment. Don't forget any adapters and chargers you require. One of the most valuable additions to my children's carry-ons is their own mini sketch pad, pencil case, colored pencil set, and pencil sharpener.
Because planes are always chilly, I pack a small blanket that I have gotten from other planes in each of my kids’ backpacks. This way if little kids want to nap or are cold, they can make themselves comfortable. You could also add a mini pillow or travel pillow for mid-flight naps.
Our kids pack one book, one small toy if requested (such as a small car or small doll) and of course their lovies. All of these will serve to alleviate boredom while waiting in the airport and are an antidote to non-stop electronic use during the flight.