This is one of those projects.
I was able to make it with entirely recycled materials that were around my house, but even if you needed to purchase a few things, it would definitely be less than $10.
Fabric to cover and line box (a yard should do the trick unless you have an extra large box)
Adhesive that will work on fabric (I used a spray)
Knife (I just used a serrated bread knife...I guess if you want to be fancy you could use a craft knife)
Circle Template (I used a small glass dinner plate)
Trace two circles on the front of the box (or be a rebel and make it a one handed box and trace just one). Cut out the circles using the knife.
Line the bottom and sides of the box with a piece of fabric
Cut a piece of fabric the width of the box, roughly as long as the height plus the length
Glue the fabric along the top edge above the holes, leaving the rest of the fabric lose.
The fabric will hang down and cover the holes, make sure the fabric is hanging pretty side out.
Tape the box shut.
Wrap the entire box in fabric like a present.
You may need to do some trimming to keep it from getting bulky.
Cut out the circular openings and glue the fabric back around the cardboard edges.
This is a tricky step, so take your time!
Ta-da- a free or very cheap mystery box!
Mystery Box Ideas
Basic- Guess items that are in the box from touch only.
Intermediate- Place several items in the box and have the child "find" a specific thing (maybe some letters from the moveable alphabet?)
Advanced- Draw the item in the box without looking at it
Expert- Describe the item in a way that another person can draw the item without looking OR touching!