When you were born, the medicine man or woman would preform a ceremony "i.e. Today's traditional naming ceremony in other cultures) and they would deem the necessary name, depending on what type of personality they saw in you. It may have been simple and as you grew they added on it or in some tribes you didn't have a name at all til you were a little older (around the age of 2 or 3 you were just called 'son' or 'daughter'). Warriors usually had their named changed after a great battle or feat, or had something added on to their original name. If you were adopted or married into the tribe and didn't already have an indian name, you would be given a name by your adopted mother or the medicine man or woman by how you acted, your personality, or something that just stood out to them (i.e. A eagle followed you into camp, you have the eyes of a wolf, you have the soul of a mother bear, etc etc). This is still done today in many tribes, and now some Indians today just name their children or themselves without the help of a medicine man or woman, or just have the medicine person there to bless the name.
The cherokee indians got their names after contacting their spiritual animal spirits, for instance, they would go to a certain place, or in a teepee, they would smoke a certain thing in that pipe, an animal would appear to them. Also, as a young child, they would send you out into the wilderness to come back with food for the tribe. If you came back with any kind of food, you were a warrior!