The fictional gift-giver in Tolkien’s published letters to his children is Father Christmas.
Tolkien wrote illustrated letters to his children over many years, presenting them as messages from Father Christmas at the North Pole. The letters describe imagined events, helpers, mishaps, and battles, and they include drawings and invented writing systems.
The collection was published posthumously as The Father Christmas Letters. It is separate from Tolkien’s Middle-earth books, although it shows the same delight in languages, maps, detailed histories, and layered fictional voices. The letters began in 1920, when Tolkien’s eldest child, John, was three years old, and continued until Tolkien’s children were older.