Memory work has a reputation problem in children's ministry. Most of us remember it as recitation under pressure, and a lot of volunteers quietly drop it because they do not want to be the adult who makes a child feel stupid in front of the room. The fix we found was not to drop it but to change the shape of it: pick verses that have some internal structure, and let that structure become a game.
Anything that names several items in order can be played as a sorting game. Ephesians 6 and the pieces of armor is the obvious one, and the Fruit of the Spirit in Galatians 5 works the same way. Write the items on cards, shuffle them, and let a team put them back in order before anyone reads the verse. They will get the order wrong. That is the point: the reading afterwards is the correction, and children remember corrections they made themselves.
Proverbs is full of two-part sayings where the second half turns against the first. Cut the halves apart, mix them with halves from three other proverbs, and let a group rebuild them. It teaches the shape of Hebrew poetry without anyone having to say the words "Hebrew poetry" out loud.
Five loaves, two fish, twelve baskets, forty days, seventy times seven. Numbers are the easiest thing in the world to build quiz rounds around, and children who cannot yet articulate the meaning of a passage can hold onto the numbers and use them as handles later.
If your congregation reads an older translation, the vocabulary itself is worth a game. Crosswords are unbeatable for this. We keep a stack of printed grids in the cupboard for evenings that need a quiet twenty minutes, and we pull ready-made ones from bible4soul's Bible crossword when nobody has time to build a grid by hand. A child who has spelled "raiment" into a grid will not stumble over it a week later in the reading.
None of this is complicated, and none of it requires a budget. It requires deciding that the goal is a child who still has the verse in October, rather than a child who performed it correctly in September.