After much tooling around, I discovered that Flash itself does not provide the ability to automatically set a quality rating. It's done by the author of the movie or animation, who usually set the quality to HIGH. But, with the power of Firefox and an add-in called Greasemonkey, you can set Firefox to automatically set the Flash quality to LOW.

This works great for older PCs and Macs or people with very slow Internet connections. It is also ideal for children with a One Laptop Per Child computer (OLPC) who have installed Firefox.

Step 1: Install Greasemonkey

First, with Firefox, download the Greasemonkey add-on here. Install the add-on, then restart Firefox. Then return to this article to finish the tutorial.

Step 2: Install the User Script

Next, you'll need to install a user script called Force Low Quality Flash. The credit for this goes to a user by the name of Mikado who posted the solution in a Greasemonkey forum. I made the script available for public download here. When you go to that link, click the Install button and it will grab the latest version of the script.

Step 3: Test

Next, go to a site heavy on Flash. My children frequent Noggin.com, so that's where I go to test it out.  After the Flash animation loads, right-click on it and check the quality. If it is set to Low, we have success!