With my budgies, I just came up with whatever name I thought might suit them (except for Rumi, Lara, and Pollo, who came with names, and I really didn't feel like renaming them).
Tiki and Ria were already named, but their names were stupid, and I wanted to give them Maori names in honour of their heritage. It turns out most Maori names on the internet suck (they're just names of random gods, nature spirits, mountains, etc.), but I managed to find the names Tiki and Ria.
As for Noah, I wanted to give him an actual name (no way was he going to keep his original name, "Thor"). I spent forever running names by Mom, when she suddenly suggested the name "Noah".
With my hermit crabs (not parrots), I went with a nature theme to try to give them some sort of dignity, given they were ripped from the wild and suffered terrible abuse in captivity. I adopted all five of them, and I gave them the names Oliver, Lily (she's pink and delicate like a water lily), Kai, Kilimanjaro, and Percy. Percy definitely isn't a nature name, but I decided to give up on the theme at that point.