Use zippered pencil pouches to organize puzzles instead of storing the bulky boxes. Cut off the image from the front of the box and store it in the bag, too. (This blog post has several great ideas for organizing kids toys.)
Buy a Bucket Boss (a portable tool belt that hooks on a bucket) from a building supply store for about $7, put it on a bucket (duh), and add the kids' art supplies & small games. Inside holds big stuff like paper and all the pockets on the outside are for pens, markers, glue, scissors, small games, etc.