Jacob Tu Tu's father was right on the money when he said "do it until you get it right, then keep doing it until you can't get it wrong." He inspired me to keep playing the hymn I was learning on the piano. Even when I couldn't get to it for several weeks, it was always in the back of my mind, and when I sat down at the piano to play I practiced the same hymn. Many times it was boring, yes. And many times I just wanted to start playing something else. But I was determined to memorize this hymn if it was the last thing I did.
Well, I am happy to report success. Not perfection, mind you. But I can play it many times through without a mistake, and how it thrills my soul to be able to worship the Lord without distraction on the piano. Another benefit is that I can work on technique without having to worry about which keys to press. It is so freeing! Also, I have already noticed that playing other songs is easier. Maybe because my brain is more easily recognizing common intervals. For the record, my goal is not to become a concert pianist, but merely to be able to easily play hymns and perhaps to compose simple worship songs.
Memorizing scripture gives the same benefits. When we take the time to commit His word to our minds, it is easier to focus on what He is saying to us. It is also easier to understand His word, for the scriptures interpret themselves when you have other related scriptures coming to mind as you read. When you are well-nourished, it is easier to enjoy the presence of the Lord :-).
Much like the habits in the physical world bring a certain automation to our activities, memorizing written material brings automation to our minds, and memorizing music brings an ease to playing heretofore unknown. It is like a groove which is whittled deeper each time you pass over it. Soon it is no longer a chore to move the same way. Eventually it is difficult to go any other way. You are hemmed in and able to enjoy the activity.
I am convinced that mentally there is really no difference between memorizing a song on the piano, and learning discipline (or teaching it to a child). Until a person is constrained to obedience and self-control, he is really not free to enjoy his life (or activity). Just as a young child who is being allowed to disobey or break rules is unable to be truly happy for the mental tension which is present in his little heart, so the child of God is unable to be truly happy if God is not the ruler of his spirit. Indeed he will not be happy until he is hemmed in by the author and perfecter of our faith; walking in the Spirit with all diligence.