Memorizing poems, songs and rhymes seems to be a bit easier for kids and adults because they usually have some sort of rhythmic pattern or tune. Start by only memorizing a sentence at a time. Work on memorizing the poem, song, rhyme for no more than 15 minutes or so. Any longer may just create frustration for both you and your child.
Start with a poem, song, rhyme that has about four lines. Focus on the rhythmic pattern. If it doesn’t already have one, create it. After a while you can work on memorizing longer poems, songs, or rhymes.