My mom always did a great job of helping us tuck God's word away in our minds. We memorized Psalms 23, the Lord's Prayer, part of Psalms 119 and more. Mom knew that young children could memorize so much easier than adults could. That was such a foundational part of my growing up and I am so grateful for it.
When I went to Teen Missions Int'l ('92,'93, '94), I had to memorize more verses. The first year, I had to memorize 40 verses - almost one a day. In subsequent years, I memorized verses on either side of the original 40. To this day, there are situations that will arise and some of those verses pop right up in my mind. I think that is part of what the Scriptures talk about when they say to be "ready in season and out of season".
Kids are able to learn way more than adults give them credit for. They may not understand all of it, but the verses are there and ready when they need them. We have been memorizing all kinds of Proverbs and some select New Testament verses for close to 2 years now and I decided to take on a challenge. I read about a family that encourages their children to use Matthew 18 as a biblical model for conflict resolution, so I decided to join them. We started learning Mat.18 this summer and, while we haven't been as regular as I had planned, we have been doing it in bits and pieces off and on. I keep a copy of Mat.18 in the car so that when we are just sitting and waiting, we pull it out and work on our verses. I also review them with the girls at mealtimes when I am waiting for them to finish eating :) I strongly encourage you, if you have young children, to start memorizing with them. Hand motions help a lot, but they can learn faster than you probably can!
We are up to Mat. 18:14, but the last 4 verses are a bit more challenging and we just learned them this week, so Elianna wasn't ready to try them yet. So, here is Elianna with Mat. 18:1-10 and Janaya with verses 1-4 or so. Enjoy!