I had a professor once say that Native American tribe names were basically neighboring tribes saying "Those guys over there" in their own language. Was never sure if this was actually the case.
That is true, most tribe names known by non-Indians are not titles originally used by Indians themselves.
Members of the Blackfeet Confederacy are believed to have been given the nickname "Blackfeet" by outsiders referring to the dark bottoms of their moccasins. The Blackfeet referred to themselves as Niitzitapi, which simply means "the people." The Blackfeet Confederacy was a collection of several tribes (Piegan, Siksika, and Kaina) sharing similar culture who treated with the United States at the same time. Though on the present-day American reservation (there's a Canadian reservation too) these past divisions have fallen out of use and most identify solely as Blackfeet. Why is this? Because people change over time and new experiences and trials can serve to divide and unite people in new ways.
Many Indians were removed from the Columbia River during the damming carried out by the U.S. government onto reservations like Colville, Yakama and Umatilla (there are others too). These people do not just identify by their government-named reservation, but see themselves as "Columbia River People," as they had been fishing and trading on the river for centuries. But their division from the river lead to new identity-shaping events.
The Navajo, a tribe living in northern New Mexico, get their name from a mix between a Spanish and Pueblo word. The Navajos actually call themselves the Diné, which means "the people." Sioux is a Chippewa slang (means something like raiders/serpents/enemy) that the U.S. government used to describe the Lakota, Dakota and Nakota peoples. Whereas Lakota, the name used by the native people, actually means "dwellers on the prairie."
In British Columbia, First Nations people around the Fraser River call themselves Stó:lō, which means "river people." However, before they reached the River their mythology called themselves Tel Swayel or Tel Temexw, which means "sky people" and "earth people."
During the Red Power Civil Rights movement, different Native American groups worked to create a pan-Indian identity. They spoke of a broader Indian experience of broken treaties, stolen lands, and a need for greater self-determination. Indians were able to use this new identify to gain recognition on a national and international level.
I've tried to include a long list of different tribe names to show how identity is a malleable thing. American and Canadian Indian peoples have identified themselves differently through time time for a variety of reasons. Some have taken on names used by outsiders (both white and Indian), had names placed on them during drastic change, and others changed their conception of identity on their own terms.
SOURCES
The Reservation Blackfeet by William E. Farr
Shadow Tribe by Andrew H. Fisher
The Power of Place and the Problem of Time by Keith Thor Carlson
Blood and Thunder by Hampton Sides
For the Aztecs they actually called themselves Mexica which means "the Mexica, or the Mexicas, the people of Mexico (plural of mexicatl)". And Mexico is merely the suffix for place (-co) added to the name of people (Mexica).
For the Cherokee, the name they had for themselves was "Tsa-La-Gi" (ᏣᎳᎩ), which translated means "Principal People". I'm not sure what they called other tribes.
Toponyms to identify people groups have a long history in Mesoamerica, but modern names typically reflect non
As is usual, the Maya left us the most complete record of names, but other cultures did have their own conventions. Of the few Zapotec symbols from Oaxaca that appear to have a glyphic significance, some have been identified as toponyms indicating where a person was from. In central Mexico, the grand city of Teotihuacan appears to have used similar methods. The earliest symbols appear to indicate places by key geographic features. This abstracted into using terms like "hill" or "mountain" almost to mean "place," something the Maya may have picked up. Given out limited knowledge of Teotihuacan writing, it can be difficult to tell if a motif represents a person, people, place, or deity. This is the case with "Spearthrower Owl Hill". Its use in Teotihuacan suggests it is a toponym, possibly refering to the Temple of the Moon at the city center. But at the Maya city Tikal, inscriptions describe either the Spearthrower Owl people (Teotihucanos) or a man named Spearthrower Owl who supported a usurper in the 4th century. However, the modern name for the city is actually borrowed from the Aztecs.
For distinguishing their own peoples, the Maya used emblem glyphs. Each major city-state/dynasty had their own glyph which identified themselves. You can find a selection of them here (pg. 11). Some were concrete: the people of Kalakmul went by kaan, "snake." Others' original meanings, if there were any, are lost: the people of what we call Piedras Negras were once y'okib, for which we have no translation. Others are abstract: rulers of Yaxchilan called it pa'chan, "split sky." The glyphs are most often used as titles for figures in a text or to record the conquest of a foreign people. David Stuart and Stephen Houston have a good discussion of it here, though it is a little dated.
The original Five Nations of the Iroquois Confederacy sort of follow this trend. We all refer to ourselves as Ögwe'o:wëh which like other Native American peoples means "Real People" but when we identify as individual nations we say the names we gave each other. The Senecas are called Onöndowa'ga:' or "People of the Great Hill". The actual translation is just "great hill", the people and specific hill are implied. The name "Seneca" is a mispronounciation of an old Mohawk word that they used to refer to the rest of the nations in the confederacy. Cayuga is a mispronounciation which means "People of the Swamp", Onondaga means "People of the Hills", and Oneida are "People of the Standing Stone". Mohawks however refer to themselves as "People of the Flint" this doesn't have to much to do with their location as it does trade. The word Mohawk is theorized to be Algonquin or Abinaki for "man eaters", this may have been also used to refer to the confederacy as a whole by their eastern enemies. Most of the names for our neighbors are forgotten. However the Delaware and Meskwaki, for example we have names for but don't know the exact translation. I know that Cherokees are called Oyáda'gé:a' which means "Cave Dwellers", why they are called this I'm unsure. The more neighboring tribes became absorbed and exstiguished the less we used their names. There were so many Delaware people absorbed into villages that they almost became the Seventh nation in the now Six nations of the confederacy. Some families, although on paper are legally Iroquois in the eyes of the government, even identify as Delaware. These days we use Ögwe'o:wëh to refer to any Native American person, the way the elders use it makes me think that it has been used this way long before the pan-Indian movement of the 60's and 70's but not anytime before the reservation system. That's what I heard anyways