Fellowship of the Ring closed with Enya’s “May It Be.” The Two Towers closed with “Gollum’s Song” by Emiliana Torrini.
Return of the King should close with “Frodo of the Nine Fingers.”
It doesn’t have to be sung by Aerosmith, though I appreciated their take on the classic Spider-Man theme song. But for people of a certain age, fond memories are had of the classic Rankin-Bass animated adaptations of The Hobbit and Return of the King, and part of what made those adaptations so memorable are the catchy songs. “Frodo of the Nine Fingers” sums up Tolkien’s saga so well, that not using Glenn Yarbrough’s song to close out the trilogy seems wrong.