Compose on piano or guitar. These instruments are versatile; they can be played in any key, and they are popular sing-along (accompaniment) instruments. Children's voices are best suited to the key of C, which is easily accessible on piano and guitar. Children's songs are almost always in major keys.
Pick your message. Children's songs can be silly and absurd, they can share an important message, and can even teach a lesson. Some teachers, for example, explain scientific principles through song, or use songs to make it easier for children to memorize long lists of items.
Pick your melody. Children's songs work on simple scales. They're catchy and easy to sing. If you're having trouble with a melody, consider parody. Melodies can be re-used to great effect; "Twinkle, Twinkle, Little Star" and "The Alphabet Song" have the same melody--a fact some people don't realize until adulthood. See Reference 1 for a list of popular melodies in the copyright-free public domain.
Pick your tempo and feel. In general, children's songs are upbeat. Play them faster than you think you should; children's resting heartbeats--where you get your basic sense of rhythm--are higher than those of adults. What feels just right to you may drag to them. As for feel, play with dynamics. For example, write a section that grows quiet, even to a whisper, then gets loud. If you want an idea of how much children love this, watch a kid dance to "Shout" at a wedding.
Write for participation. Many classic children's songs are toys as well--they offer children a chance to play or sing along. Think of the finger-walking of "The Itsy Bitsy Spider," the clapping in "Bingo," the animal noises in "Old MacDonald" and the sound effects of "The Wheels On The Bus."