Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
Mythological names, such as Firebird, Phoenix, Sphinx and Thunderbird, are excellent options for exotic birds. Uncommon mythological bird names include Garuda, Griffin, Habrol, Samjoko, Harpies, Sampatti, Ziz, Jatayu and Bennu. Some people choose to name their birds according to their coloration.