My son started backpacking at around age 5, carrying an REI comet. It was really to big for him but he WANTED to do it. My daughter, who has been backpacking since she was six (now 8) just got her own real backpack. Previously, she was using a school bag, and I had to carry her sleeping bag. Now she has an Osprey Jib which fits her just fine and allows for plenty of grow room. Osprey lists an even smaller child's backpack on their site but I haven't seen it anywhere. For trips with just one of them I take:

Clip flashlight
Noah's Tarp if there is rain predicted
alcohol stove
pot and fry pan (never understimate the power of pancakes with kids!)
water purifier
dromedary bag
bear bag
first aid kit
trowel and toilet paper
the usual stuff like matches, sunscreen etc..
my own gear and the food.
I usually end up around 30 lbs

The child (which ever one for the trip) take:
their own
sleeping bag (TNF Tigger)
rain gear
clothes- fleece, nylon and wool!
water bottle
cup and spoon
headlamp
a soft cover book
games like uno, cards, or yahtzee

My goal is to keep their backpack weight under 1/5 of their weight-not an easy thing to do when they are young and weigh less than 50 lbs.