I learned this little memory trick many years ago, and I still use it to remember the order of the books in the Book of Mormon. The Book of Mormon has 15 books, so it works well to separate the books into 3 groups of 5. Children love finger games, so use the movement of your fingers and hands to go along with the clues.
(hand one) 1-2-J-E-J
(hand two) O-We-Made-A-Hand
(hand three) 3-4-M-E-M
Here are the books listed in order along with the memory clue:
1 - 1 Nephi
J - Jacob
J - Jarom
A - Alma
Spend some time talking about the clues. You can ask your children some of these questions or make up your own questions. What's the same about group one and three? What's different about group one and three? How can you remember the J's go with 1&2 and the M's go with 3&4? How does your hand help you remember group two?
Here's one idea to do with children who are learning to read and write. Have them trace their hand 3 times on a sheet of paper. Then have them write the names of each book on the fingers.
They can color and decorate their paper. Then have them put it in the front of their Book of Mormon scrapbook.
Here are some sheets we've created. There are three different versions. You are welcome to use any of these if they are helpful.