In order to recall the words of God while we are doing other tasks, we have to have it in our minds in the first place. That is the purpose of memorizing Scripture—so that we can "meditate on it day and night" (Psalm 1:2, Joshua 1:8) Meditating on God's Word renews our minds, and renewing our minds transforms us—that's what Romans 12:2 tells us. We need to learn to think differently.
II Corinthians 10:4-5 makes us aware that we're in a war. That war is fought in our minds: we have to pull down strongholds of wrong thinking and cast down arguments and any other high-minded thing that exalts itself against the knowledge of God. Every thought is to be brought into captivity and made subject to and obedient to Christ. Paul is talking about having a proper and true worldview.
A worldview is not some philosophic impracticality that is primarily relegated to college classrooms. Our worldview determines everything about us. It is the basis of all our decision-making whether we realize it or not. The world around us in the form of media, politics, and entertainment constantly broadcasts its worldview. I don't think we ever fully perceive how much our worldview is shaped by our culture. But our worldview is also shaped—or should I say, warped—by our own sin nature, our inherent propensity to selfishness and putting I in the center of every thought and decision. There is only one ray of hope for re-orienting our worldview on the truth, and that is the Word of God. In order to conform our thinking to the truth, we can't just gain a cursory head knowledge. We need that truth to soak into every fiber of our being by saturating ourselves day and night in the Bible, God's written Word. Saturation in the Word of God leads to conformity to Truth, which in turn produces transformation. This absorption and ingestion of Truth comes through one method: meditation. It is meditation that transfers head knowledge to the heart and makes us new, different from what we once were. As stated in the opening paragraph, meditation begins with memorization. Memorization is simply a means to the end, not the end itself. In order to memorize effectively, it helps to use some kind of memory hook. Listed below are some memory-hook ideas: Memorize a verse that starts with each letter of the alphabet, then start over and learn a second set!
Memorize a verse for every attribute of God.
Memorize a set of verses that explain the Gospel message.
Choose a topic, look up verses on that topic and memorize them.
Are you struggling with something? Look up verses on that topic to memorize (e.g., anger, anxiety, pride, lust, distrust, or unforgiveness).
What worldview issues are your dealing with? Memorize verses that target those topics (e.g., evolution vs. creation, abortion vs. pro-life, etc.).
Memorize a whole passage, chapter, or book of the Bible. (e.g., The Sermon on the Mount in Matthew 5-7 or just the Beatitudes; all the Psalms that are 8 verses or less—you could memorize the whole of Psalm 119, one eight-verse section at a time. That will give you a high view of God's Word! Or try a short book like Titus or I John, etc.)
I've been trying off and on over the past few years to memorize the book of Romans. I've not been successful yet in being able to really lock it into my memory so that I could say it, but I keep plugging away at it. However, just from going over it again and again, by continually coming back to it and trying some more, by praying over it and asking God for insight and understanding into its full meaning, I am learning and growing in the knowledge of God and of Christ. As Peter exhorts us in II Peter 3:18: "But grow in grace in the knowledge of our Lord and Savior, Jesus Christ." Anyone want to join me in this battle to conquer Romans?? That verse in II Peter reminds me of one more thing regarding memorization. I learned that verse as a child. As I get older, memorization gets more and more difficult—that is, I may be able to memorize a verse rather quickly, but it doesn't stick very well. But the verses I learned as child are still with me. Parents of young children, I exhort you to fill your children's minds with Scripture. I could not emphasize this too strongly. The primary thing I wish I could go back and do over would be to spend more time and energy helping my children pack away Bible memory verses into their fresh, young minds. Remember again this progression: memorization ➔ meditation ➔ obedience ➔ transformation.