A rubber band, 2 popsicle sticks (tongue depressors), two 2"X3" pieces of construction paper or card stock, and some tape are all that is required for a child to make a simple kazoo.
The popsicle sticks can be decorated with colored tape or sharpies. Don't use regular markers that will run when they get wet and color the young musician's mouth. Stretch a thick rubber band around one of the popsicle sticks. Then fold each piece of paper 3 times (alternating direction), and tape the paper to the ends of the other stick. Sandwich the two sticks with the paper in-between and wrap tape around the outside two ends.
Hold the kazoo to your mouth like a harmonica and blow. It will sound like a little party horn.
The giggles that come from my students once they figure out how to work their kazoos are even better than the sound from the instrument.
Be sure to have enough supplies on hand for your child to experiment with various widths of rubber bands and thicknesses of paper to get the best sound.