The Preamble is the first sentence of the Constitution, beginning with "We the People," making it easy to remember. One game might be a memorization contest. The Preamble is 52 words, so for every correct word the kid gets a point. It allows kids to learn a few big words in the process, such as "tranquility" and "posterity." Before the contest, carefully explain what all the words mean so that they paint pictures in the mind. Quiz them on the definitions, but be sensitive to the fact some of the words are not commonly used even by adults. The object is to get kids to be able to recite the Preamble like the Pledge of Allegiance.
There are seven Articles after the Preamble. Try to keep the game simple. One way to do this, as in the Preamble suggestion, is stick with definitions and award points for accuracy. For example, if they name Article One as "Legislative power" and they mention Congress is made up of Senators and Representatives, award them a point. It is not difficult to remember the seven Articles and what they mean, but keep in mind all of it needs to be explained in detail by an adult who can translate it into language that is familiar to kids. Simply telling them to read up on it without defining uncommon words is how it's been done for decades, and that method doesn't produce the best results. Making it fun, however, has a chance of triggering a deeper familiarity.
The first 10 of 27 amendments represent the Bill of Rights. Award points for correct answers as to what each amendment is about. Keep it simple. For example, the First Amendment is freedom of expression, the Second is the right to bear arms. To mix it up, you can put 27 numbers in a hat and kids have to guess what amendment corresponds with the correct number. The point total for everything so far is 86. Think of a creative way to come up with 14 bonus points to make the grand total 100. Then have an election between the top two students to see who gets to be class President.