Thanks for the comments, everyone! Here's the promised tutorial!
You will need:
Glue (lots and lots of it)
First, cut 11 rectangles (mine measure 11cm x 6cm) out of the foamboard. If you want your box to be taller, you can cut a couple more rectangles.
On one of the rectangles, roughly draw with a pencil how big you want the hole to be. Then measure with a ruler (again, mine measures 8cm x 4cm), and cut a cardbard template.
Trace the opening shape onto all your rectangles, into a slightly different position on each one.
Grab your utility knife and trim out the center openings. DON'T cut an opening on two of the rectangles - those will be your lid and bottom of the box.
Now, the fun begins! Bring out the paper! I am using this gorgeous wrapping paper I just bought!
Cut 9 pieces of paper, all slightly larger than your rectangles. For the lid and bottom, cut two pieces of paper that are twice the size of your rectangles.
Now glue your rectangles on the wrong side of the paper, then wrap and glue them so they look like that. Also wrap your lid and bottom of the box.
Now, with your utility knife cut out the paper covering the opening.
Begin to glue your rectangles together in a stack. Lay each one on top of the next, making sure the openings are aligned. The outer edges will be irregular.
All done! Paint the inside walls of the opening with glue and let dry.
Now cut a lining from your paper to fit the sides of your box, and glue it into place.
Now glue the bottom of your box - you should align the outer edges of it with the outer edges of the last rectangle of the stack.
Cut a piece of paper the size of your original rectangles, and glue it on the top rectangle to cover the edges of the lining. Cut out the center with your utility knife.
For the hinges, cut two wide strips of paper, and fold them in three, so that they are sturdier and more narrow. Glue the edges on the inside of the lid, then glue the remaining strips on the top rectangle of your stuck. If you want, secure your hinges with a bit of tape.
If anyone makes a box using this tutorial, I'd love to see pictures of it!