I found the greatest dice storage option at the dollar store.
Little Lego boxes!
They're kind of small for actual Legos, but these party favors are just the right size for a set of dice.
The lid clips closed to keep them inside and the boxes stack just like Legos. Go check out the party supply section at your local dollar store. They come four to a pack in red, yellow, blue and green. Perhaps you can find other colors.