When I was a child my favorite children's story was called Mr. Tickle Go. Besides Mr. Tickle go himself there were many other characters such as the post man and fireman, his close friends. The basic plot revolved around Mr. Tickle Go's trips to different places and the people her would encounter. I remember parts of the story rhyming, but not all of them. In children's stories the subjects usually touched by authors are about families and friends.
Some authors attempt to teach children a lesson or moral.
I remeber reading the Berenstein Bears books and those always had a lesson to learn at the end. Some are written purely for entertainment value like those Pokemon stories. In general children's stories use interesting characters to convey their meaning. I think that successful children's stories have to be very original. Peter Pan is a classic story because of its very creative nature.