This is a good sit-down project for a sunny day.
First step in paper-making project
Begin by getting a sheet of newspaper, and let them cut or tear it into small pieces, about 1/2 inch square. Then soak the pieces in water, until they are very squishy. Stir them around so they totally disintegrate into sludge. You can make the paper different colors by adding a drop of food coloring to the water.
Second step: window-screens
Now, take window-screens (or small pieces of window-screen) and dip them into the newspaper sludge until they are covered, and carefully lift them back out again. Set the screens in the sun until the paper is dry, and then you should be able to peel it off the screen.
For further discussion
That’s basically all there is to making paper. You may want to discuss whether this is how paper is made today, and how the modern process is the same or different (can they find out using the Internet?). It would be interesting to calculate the area of the newspaper sheet they each started with, and the area of the paper they made with it – which is bigger? Why? Is there any difference in the weight?
You might also investigate whether pens will write well on their paper, and try adding things to the water to make the paper easier to write on. One common additive is powdered chalk.