Proverbs 9:10 - The fear of the LORD is the beginning of wisdom, and knowledge of the Holy One is understanding.
Memorizing what God says is commanded in the Bible. God told Joshua directly in Joshua 1:8 "Keep this Book of the Law always on your lips; meditate on it day and night, so that you may be careful to do everything written in it. Then you will be prosperous and successful."
To meditate on something day and night means to think about it all of the time. Constantly. Consistently. Unfailingly. You cannot meditate on something if you don't have at least some of it memorized. A Christian must deliberately meditate on the Word of God in order to "be careful to do everything written in it."
If Scripture memorization is so essential for Christ followers, then doesn't that inevitably mean that we are also to demonstrate this to our children, and help them to remember His Word (Prov 22:6)?
It is vital that young Christians have key passages for trials in your life. During times of temptation, testing, anxiety, or trouble it is ideal to have Scripture already hidden in your heart (Psalm 119:11) because you may not have a Bible on hand. If God’s word is already implanted in your heart, the Holy Spirit will use it during those times, bring it to your mind, and bring you comfort.
Certainly, if the habit of memorization is established very young, a child is given a heritage that will help him considerably in future memorizing. Many adults claim that it is difficult for them to memorize. Whether that is true in every case is open to question. It may be that they have never been taught to memorize – either at home or school.
Paraphrased from Teaching Bible Memory – Rod and Staff Publishers, Inc.