So, I did a bit of research. I learned that the inside bladder of the ball is made from the kind of rubber you have in a bike tire inner tube. This instructable will show you how I successfully patched the ball.
Step 1: Gather Your Materials
1 punctured ball
some gritty sandpaper
some fine sandpaper
small, pointy scissors
a bicycle inner tube patch kit (adhesive and patch)
some hot pink Duc tape (not pictured)
Step 2: Expose the Inner Bladder of the Ball Around the Hole
Eventually, you will be able to pull the outer skin apart from the bladder and use the small scissors to cut it away.
Step 3: Apply the Patch
The patch will not stick to the outer skin of the ball, and the edges will be wont to come peeling off, so you should trim as much excess patch as you can and then cover the patch with Duc tape to protect it and keep it flat.
Voilà! You may now pump that baby up and get back to your game.