Ooh, I would have liked to name my bird something majestic and pretty, probably something that corresponds to its color. I keep thinking about the
Eragon books, where the boy named his dragon
Saphira, after her blue (Sapphire) scales. I would have done something like that. Because Link's Loftwing is crimson, I'd have probably used a name that goes along with red. I don't actually know many names that go along with red, so I decided to use good ol' Google to come up with a few ideas.
Coquelicot: "brilliant red" - This name is rather outlandish and not very graceful, so I wouldn't actually use it. I like the meaning of "brilliant red," though.
Flammeous: "flame-colored" - The names "Flame" or "Blaze" would have been very, very cool.
Haematic: "blood-colored" - The meaning of the name is epic, though I'd probably alter it to something like "Haema."
Piceous: "like pitch; inflammable; reddish black" - "Inflammable" is a pretty cool meaning to have behind a name.
Although, if I had my way, I'd make Link's bird blue and name her/him Saphira/Cobalt (depending on whatever gender I make it). Like many of you here, I'm sure, I might choose to name Link's Loftwing something lewd of funny just because.