I sent a friend request on Facebook about a year ago that was denied and the Add Friend Button no longer showed next to their name. Recently the Add Friend option appeared again next to their name. Does it appear on its own after a certain amount of time or does the user have to allow it to appear?
In my experience the "add friend" disappears if the person deletes/rejects your friend request. I tested this with a dummy account. Some of the people I sent requests that were rejected the "add friend" after a period of time returns. However if that person marks you "spam" the "add friend" button never returns. I do not know how long the "add friend" returns but I suspect that it is a period of time that Facebook determines generically.