In the case of Legolas, his father's name is used a couple of times. The main reason we don't see it more is that he isn't introduced or asked his identity nearly as much as Frodo.
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'Welcome son of Thranduil! Too seldom do my kindred journey hither from the North.'
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He was named Elf-friend because of the great love that grew between him and Legolas, son of King Thranduil, and because of his reverence for the Lady Galadriel.
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