We've used several books. When we were wanting shorter passages, it usually meant we were doing a topical study. So the following books have Bible verses that go with a topic that is studied.
Leading Little Ones to God by Marian M. Schoolland is an oldie but goodie. It has 86 short lessons covering topics like: God is Very Great, Sin Spoiled the World, What Jesus Did, How We Become Children of God, About God's Church .... Each lesson also gives questions to ask the child, a memory verse, suggested Bible reading, a hymn (just the lyrics) and a prayer. The only downside is that there aren't lots of pictures. But if your little one is used to listening to stories, it would be fine.
A Faith to Grow On by John MacArthur has about 80 lessons very short lessons that cover the basic doctrines (God, Creation, Sin, Bible, Jesus, Forgiveness ...). Each lesson has a suggested activity to go along with the lesson, a verse (or two) that the child could memorize (some are a few verses long, so you might have to choose a shorter section of it), a prayer starter, an application question and a word scramble puzzle (which we skipped with the children weren't reading or writing yet)
Sword Fighting by Karyn Henley is a 52 week devotional that centers around character building. It is centered around a memory verse and also has aplication games to help cement the principle. It says it is geared towrad elementary students, but my younger ones weren't in school yet when they used it.
We've also gone through the AWANA program that has kids memorizing lots of verses. (The program itself includes a kid level Bible study and a game time.) They start with partial verses in the 2 years of pre-K (I think it is 3 & 4 year olds), then move to whole verses. I'm not sure they sell just the books, but there might be a church near you that can show you what they use. The
www.AWANA.org website has a "Find Awana in your area" button. They also have music CDs that go with the verses to help the children memorize the verses.
The other thing my boys liked was Steve Green's
Hide Em in your Heart CDs. I think he has 2 volumes. It is scripture put to music with a little commentary to help them apply the verse.
I'm sure I'd be able to come up with more
but I should probably stop there.