I just finished this project for my son. It is very easy to make and cost almost nothing. All I used was things that would normally be thrown in the household garbage and things that I already had. No spending required.
My son loves his new board and will peek-a-boo over and over again.
Below I have the instructions on how to make this board as well as a list of the materials needed. 
Homemade Peek-A-Boo Board


Materials Needed:


pen or pencil
ruler or straight edge                                                                  
scissors                                                                                                    
hot glue gun or strong tape
1 thick piece of cardboard (old packing box works great)
thin pieces of cardboard (cereal, biscuit, or empty tissue boxes)
stickers or pictures cut from magazines 
(pics of babies, people or animals work great for baby)
scrap decorative paper (optional)





Instructions:
1. Cover one side of the thick piece of cardboard with a piece of decorative paper. This is optional.


2. Using a ruler or straight edge start measuring the square flaps. It is better to include some of the cardboard which is already creased in the box to make it easier to lift the square flaps.



3. Cut out your square flaps. 


4. This is what your flaps will look like.



5. Using the flaps as a guide, measure and cut out windows to fit the flaps.


6. Place your stickers or magazine cut outs on the board spacing them enough in between so that your flaps and windows can fit well.


7. Arrange your windows and flaps on the board and glue/tape into place.
* Only glue the top part of the flap as the two sides and bottom need to remain free in order to flip them up.


8. All finished.