Here's some recommendations. I'll update it more later, as I'm currently sick with the flu so can't stay on the computer long. For those that live in the states, all of these books and series can be purchased online at kinokuniya, which has very reasonable shipping fees in comparison to other places since they're shipping from their bookstores in the US.
10分で読めるお話 - Compilation of short stories from across the world, including fairy tales, myths, parables, and others. Each story is short enough to read in ten minutes, thus the title of the series. These are graded readers, designed to be reading supplements for elementary school children working on their reading skills. It goes from 1st grade to 6th grade, with two books per grade. The lower grades contain less kanji, large print, simple vocabulary and grammar, and then progress grade by grade from there.
なぜ?どうして?科学のお話 - By the same publisher as 10分で読めるお話, these are graded readers designed to teach elementary school kids science. They explain basic science facts, such as why skin becomes pruney in hot water, and can be informative even for adults.
魔女の宅急便 - All of you have probably seen Kiki's Delivery Service, and the novel it's based on is fantastic for learners: it uses simple grammar and vocabulary, and is a great book to read if you're just starting on Japanese books. The plot is completely different from the film as well, so it's interesting to compare the two. If you enjoy the novel, there are five sequels at about the same difficulty.
民話と伝説 呪いの巻物 - A series of scary folktales and legends, split into books of world legends, and others that are just traditional Japanese tales. If you have an interest in Japanese mythology, mainly in ghost stories and the monsters, this is a fantastic series.
獣の奏者 - A popular fantasy series that also spawned manga and anime adaptations. I'll just steal Wikipedia's blunt plot summary of, "It focuses on a girl who can control the greatest of beasts as she gets involved in a war between two kingdoms."

It's one of the more difficult reads in children's literature, compared to simpler books like 魔女の宅急便, and after this series a learner should be ready to progress out of children's books.
Will post more later, when I'm not on the verge of dying