You know those plastic "to-go" containers you get from some restaurants? (my favorite Chinese place uses them as delivery containers!) I use these containers to store my puzzles. I cut off a picture of the puzzle from the original box, tape it onto the lid of the "to-go" container, and put my puzzle pieces inside.
These make for nice, neat storage, and most of my containers have lasted me a few years! I store some puzzles in Tupperwares as well...same deal. Just tape a picture of the puzzle on the lid, and good to go!
Next time I'm craving Chinese, I may just need to go buy some puzzles!