Use index cards with before and after scenes to get them to remember what's next. Index cards with the lines on them may also help with memorization. But try mixing it up and writing the lines that come afterward or prior. These should both be other lines your child has, as well as lines from other roles in the play or musical. This helps the child remember in which order to do things when it's time for the performance.
Do mini-versions of the play or musical during practice. Not only does this help with memorization, but it also helps prevent nervousness. Many times kids get stage fright because they are afraid they'll forget their lines. It won't prevent all cases, especially those unrelated. But it can certainly help. Some kids learn by performing actions. Even for those who don't, hands-on experience is beneficial.
Record them saying the lines and play it back to them. This can be a fun memorization method for kids because they get to record themselves or have you do it. When they record the lines, they are free to read from their study book or sheet. The lines can be played back while in the car or while they do other things. Auditory learners will greatly benefit from this screenplay line memorization method.
Have them write down the lines. This helps visual learners. But it also helps the brain process the information. By writing the lines for the play or musical down, a child needs to read them, as well as think about them. Repeating the process helps keep them memorized. For each succession, have the child write the lines down at least three times each and read them aloud afterward.
Be consistent and persistent with a variety of methods. Practicing often for a good length of time will prove to be beneficial. Make sure they spend ample time every day practicing their lines, using the above methods, as well as any others you can think of. No matter their learning style or how the scenario plays out when they perform, they can feel confident they know their lines all around.
*I originally published a version of this via Yahoo Contributor Network