Here is how I made some simple patchwork balls last week, for my children to play with inside. They would make great new baby gifts, too! My daughter (3 yrs) enjoyed helping- choosing the fabrics and stuffing the finished balls. An older child might like to sew them up, too!
Materials:
small scraps of cotton fabric (I like to use quilter's-weight cotton best, because it's strong and less likely to fray at the seams.)
needle and thread, scissors, pins
paper and pencil
sewing machine (but you could sew the whole thing by hand, if you don't have a machine.)
stuffing- polyfill is best for small babies, but you could also use fabric scraps or cotton or wool stuffing.
optional: a small bell for a baby's ball, rice to put with the stuffing for extra weight, or fill with just rice to make a hacky sack/juggling ball.
1. First draw your pattern on a piece of paper. I drew a petal-like shape in two different sizes to make the two different sized balls. See the diagram below. Don't worry too much about the accuracy of your piece, as long as it is even it should be fine! To make the bigger ball, I used a petal shape cut to the dimensions given below. To make the smaller ball, my shape was approx. 8.5 cms long, and 3.5 cms at the widest point. You could experiment with the size you like best. Cut out the pattern and the small circle piece, and choose your fabrics.
2. Iron the pieces of cotton you will use, pin the pattern pieces on and cut out. For each ball, you will need six petal-shaped panels, and two small circle pieces. 
3. Sew the ball panels together. Place two pieces right-sides together, and pin. Sew from the top edge to the bottom, allowing a 1/4" seam. Don't worry too much about the top and bottom of the ball, because you will cover those bits up later. Continue like this until you have sewn all six panels together.
4. Now sew the last and first panel together, right sides together. Leave a gap in the centre of the seam, for turning and stuffing.
5. Turn the ball right-side out and stuff with your chosen filling.
6. Hand-sew up the opening. Next, hand-applique the fabric circles to each end of the ball, to cover the seam ends. If they are pretty tidy anyway, this might not be necessary. Alternatively, you could use fabric glue to attach felt circles. Voila, all finished!