Many rhyming games need no props. A simple one is to have one person say a word, and have your child (or group of children) name as many words as they can that rhyme with it. Another game involves changing one word in a sentence to a word that rhymes with it, creating a nonsense sentence. Then your child can name the correct word. For example, "I put a SING on my finger." (The correct word is RING.)
Play more games with simple props, such as rhyming cards. Write pairs of rhyming words on index cards and have your child match the pairs. You can use them as flash cards, or you can play "Memory" with the cards. To play rhyming "Memory," place the cards face down individually, in any arrangement. Turn over two cards at a time and try to find the rhyming pairs. For children who don't yet read well, use pictures of objects that rhyme instead with the word written on the card. Many rhyming games are available online.
Children who read well can learn about groups of words that rhyme. With a second- or third-grader, pick a common word ending (such as -ate), and have your child name as many words as he can that end with those letters (such as date, slate, state, rate). Make flashcards for common rhyming groups of words and add to your list in that group. You can also show older readers how to use a rhyming dictionary, and search rhyming books for words in a group.
Another way to learn about rhyming sounds is through reading. Choose a children's book with a lot of rhyming words, such as books by Dr. Seuss or nursery rhymes. You can ask your librarian for suggestions. As you read the book, have your child indicate every time she hears rhyming words. She can touch her nose, raise her hand or hold up a stop sign. Then stop reading and have her tell you the rhyming words.