Break up or paraphrase the scripture that you want the child to learn. Children do not have the ability to hold attention as long as adults do, so they can have difficulty in memorizing scripture if it is presented in one large chunk. Additionally, depending on their age, children may not have the capacity to understand fully what the scripture means---they may be unfamiliar with certain terms or thrown off by the syntax used. Using a paraphrased version of the scripture helps the child to memorize the scripture because it increases the amount of relevance that the scripture has to the child in terms of language.
Choose verses that deal with children or show how the verse to be memorized affects the child. Doing this can help children feel as though they are not left out when compared to adults and that the scriptural message is directed even to them. For instance, when having children memorize Luke 18:16 ("...let the little children come to me, and do not hinder them, for the kingdom of God belongs to such as these"), you can talk with the children about times they have been allowed to go to friends or parents and how it made them feel good to have that attention. This also allows for the opportunity to teach the character of Christ through the memorization assignment.
Use songs that correspond to the scripture verse to be memorized, or set the entire verse to music. Most children enjoy music, and many children learn best through listening rather than through movement or visual cues. You can find many scriptures that have been set to music, specifically within the choral genre. For instance, works such as Handel's "Messiah" are taken directly from scripture, while spirituals such as "Wade in the Water" or "Ezekiel Saw the Wheel" retell or deal with scriptural stories. Sharing these types of songs gives the opportunity for assessment outside of written reproduction. For songs with text taken verbatim from the Bible, you can have the child sing the song instead of recite the verse. You also can organize group pageants, where the children sing the songs together for parents.