It seems like the ___ son of ____ is used most often in greetings and introductions. I would say it is used as a way of identification. In Legolas's case for example, most people probably didn't know who he was, but they probably would know who Thranduil was, being that he had been king for the entire 3rd Age. (Before Thranduil it was his father Oropher, who was slain in the War of the Last Alliance.) Aragorn often used the "son of Arathorn" when introducing himself. To the hobbits in Bree "I am Aragorn son of Arathorn, and if by life or death I can save you, I will." To Éomer: "Elendil! I am Aragorn son of Arathorn...". to name a few. Someone like Galadriel or Elrond wouldn't need this form of ID because they were obviously quite well-known.
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