I have one character that is nameless, and the entire plot of the story revolves around his search for an identity.  
This boy, now nearly 16, was never named because his eccentric mother refused to pick a name until his father came home.  The father abandoned the family and never returned, and the kid is tired of being a nameless "status."  So he sets out to find his father and gets stuck in a 1970s orange hippie van with a bunch of other mildly insane relatives.   
 
 Obviously, the boy's anonymity plays a large part in the story.  His mother unofficially calls him "Child," and that is the name used on documents.  That is also the term the narration uses consistently.  Additionally, relatives often refer to him by relation - son, grandson, nephew, etc.
I use these "names" to make a point in the story.  The boy hates being a nameless "status," so he sets out to earn a real name and an identity.  What he learns is that part of your personal identity is in your relationship to others, and titles like "son" and "father" are very special indeed because of the relationships they signify.
Because you pick one name for a character and stick with it in the narration (action) of a script, dealing with the anonymity isn't difficult from the writer's standpoint.  What's going to be tricky is explaining that Child doesn't have a name and why... without being OTN...