Phoenix and Hawk can be names for any type of bird. Penelope is an exotic bird name. Jem is a cute version of Jemima, which means "little dove." Birdie can also be a cute and simple name that states the obvious
Phoenix means "deep red," but it also is the name for the mythical bird that supposedly lives for thousands of years before it finally explodes into flames and then proceeds to rise from its ashes. Hawks were traditionally hunting birds. In the 1800s, hawks were used to hunt as a form of aristocratic sport. Because hawks are large, fierce birds, naming a smaller bird Hawk can be ironically adorable.
Penelope was the wife of Odysseus who stayed faithful to him during the Trojan War, according to legend. In the Bible, Job's daughter was named Jemima and was considered the most astoundingly beautiful woman of her time. Jem can be a name for a colorful female parrot.
Birdie is an appropriate name for a bird that belongs to small children due to its simple and understandable meaning. Robin and Wren are other suitable names for birds; these names denote particular bird families, but they can be applied to other types of birds as well.