What's in a name?

I saw a license plate yesterday that said TGRLILY, obviously referring to the Indian Princess of the Piccanniny tribe in Peter Pan. It got me to thinking about how Native Americans received their names.
At boy scout summer camp a little while ago, they have an Indian based brotherhood called Mic-O-Say. In joining this I had had to perform certain rituals and put through work and at the end of all of it I was officially inducted as a member of the tribe of Mic-O-Say. I could go more into it, but that's not really the point of my post. As part of our trials we had to spend two nights sleeping and meditating out in the forest. At the end, we were supposed to choose our Mic-O-Say name. My name? Swift, Stealthy Cheetah in Nearby Meadow. It's ridiculously long, I know. Most people's names were like, Broken Arrow, Night Star, Standing Oak. Cheetahs are my favorite animal so I decided to go along that route. I decided I might change my name and I got questioned about it quite extensively because apparently, at the end of our meditation, once we realized our purpose in life, we would see a sign, and this sign was an indicator into what our name should be. I didn't realize this at the time. My friend, named himself Deer Hoof, because he woke up in the night and a buck was standing right next to his sleeping bag. This was his sign, consequently his name. I decided to keep the name, but hopefully noone questions my sign too much.
It has me wondering though how real Native Americans choose their names. Is it just something that is passed down through a generation, or do they have similar experiences that lead to their naming?
In class we learned how at the boarding schools, kids would be given an English name and then adopt their father's as a last name. This is revolutionary for me because I sat next to a kid in home-ec who's name is Matt Yellowbird. This must be why Yellowbird was his last name.
And a funny point is that, these obviously aren't even real, because an Indian word for "yellow bird" would not be spelled or pronounced like that. It must be subversely English.