The idea that Frodo would have written down Legolas’ name comes straight from J.R.R. Tolkien’s texts. The book the hobbit writes in “The Return of the King” is called “The Red Book of Westmarch,” and it is essentially the in-universe version of “The Hobbit” and “The Lord of the Rings.” Bilbo, Pippin, Merry, and Sam all add to it. For example, Bilbo includes a sort of diary and compendium of his translations, while Pippin and Merry supply extraneous research.
Frodo is also a huge contributor. At the end of the “Return of the King” book, he doesn’t just write down the events of the War of the Ring. He gives the book an absurdly long name: “The Downfall of the Lord of the Rings and the Return of the King (as seen by the Little People; being the memoirs of Bilbo and Frodo of the Shire, supplemented by the accounts of their friends and the learning of the Wise.) Together with extracts from Books of Lore translated by Bilbo in Rivendell.”
In other words, Frodo organizes, assembles, and records the Red Book of Westmarch, personally adding critical passages that recount the War of the Ring — the part of the story including Legolas. He undoubtedly had to recall and write the elf’s name numerous times during this process.
