A birthday box is a way for the child to reminisce and remember "their" favorite day : )  Each year starting at birth (but could be started at any time), collect some of following things and put them in a binder or decorative box.  So how does it work exactly?  Here are a list of ideas:

make a list of who came to the party

newspaper from their birthday each year
 

a personal note from anyone that chooses to write a little note to the birthday child which can prove to be valuable as older family members pass on. 
 

any accomplishments/progress for the year (talking, walking, etc)
 

photos and/or video of each birthday
 

an empty sleeve for birthday cards
 

pictures drawn by siblings and cousins for the birthday child

P.S.  For my kids, their birthday boxes also hold special items just for that child.  For example, booties that great grandma made, hospital bands, blessing outfit, plaster paris of their hands as a baby, special mementos, etc.  I consider their boxes a time capsule and none of them will get or have access to them until their 18th birthday.  Some of the grandparents did not want to write letters each year and instead wrote a very long personal letter to the child with their testimony, hopes and expectations for that child.