"And I quote..."

Kids have an amazing capacity to memorize! We started our oldest on memorizing verses when he was about three, and he just took off. None of the short and sweet, "Jesus loves us all." that seems to be the weekly verse in Sunday School. A full verse. We made verse-time a part of the nighttime routine. I believe this is key - ROUTINE. We did a lot of "repeat after me" learning, and I'm also a big advocate for figuring out a rhythm in which you say the verse each time, and also doing some appropriate actions along with the verse (God=point up, love=hand over heart, people=spread hands around to show a multitude, etc). We've gotten pretty creative over the past few years with our motion cues. Our middle child learned the verses at the same time as the oldest one, at about 16 months. We realized he was memorizing too when Sam would occasionally get stuck with a word and Caleb would supply it for him. Wow, okay, so a year and a half isn't too young. Start them early. You'll be surprised. We sure were! Anyway, after the actions are set to the verse and done a few times together, pretty soon, all we need to do are the motion cues and the kids supply the words. Then they're doing their own motion cues and saying the words. At the beginning, we spent about a week on each verse and also spent a good amount of time talking about what the verse means. They are now (at ages 5 and 3) doing the suggested memory verses from church (targeting the kids 1st grade and up) and have memorized approximately 28 verses so far since the beginning of the year.
Now since Caleb started memorizing so early and was picking it up so well, I started him on recognizing his alphabet at just over 2. He learned them in a few months. Of course, I'll be the first to say that we have the smartest kids in the world. But I do think that this early memorization opened up their little "mind capacity" box for them at an earlier-than- normal age.
I would encourage parents of young children (or older children if you haven't started yet) to start very young. Even if it's just you repeating the verse as part of the bedtime routine, your child is hearing that, and you'll be surprised when they start vocalizing it too. An added benefit for the parent - you can stop wishing that you had the diligence/time to memorize Scripture and just do it!