The term "Indian" is a collective racial term that now has a foundation in what is essentially Pan-Indianism.

Those of us who know our nations or communities generally prefer to just go by our traditional tribal names instead of a larger label. The issue surrounding a collective term will vary in importance from person to person. One big reason, though, why "Indian" is still used is because it is now a legal term. Native Americans are cemented as "American Indians" based on the past dealings between the colonizers and natives.

For example, the various governmental departments responsible for interactions with Native American tribes are termed as such; there is the Bureau of Indian Affairs and Bureau of Indian Education. Legislation has termed tribes as "Indians". You have "The Federal Indian Trust Responsibility," the "Indian Citizenship Act," and the "Indian Child Welfare Act." Many tribes have accepted the use of this term and established themselves with it in their politics, such as by incorporating it into their official name. Examples are the Puyallup Tribe of Indians and the Nisqually Indian Tribe. There has been another push since the 90s to switch to Native American, such as with the passing of the Native American Graves Protection and Repatriation Act (NAGPRA) and other social concepts, but since a big portion of how the U.S. interacts with tribes is carried out in the court room, case law and past legislation continue to dictate how the natives are to be addressed. It isn't necessarily possible to start calling Indian reservations "Native American Reservations" because that starts to conflate the term "native" and would need to be redefined. The struggle to define the concept of "Indian Country" lasted decades and defining who is and who isn't an Indian is always difficult when considering jurisdiction and citizenship (not that the feds should be deciding that anyways, but they do in many aspects). Attempting to switch to a new term would basically flip over a whole history of defining case laws and legally binding treaties made with tribes.

The use of "Native American" saw its rise in 1960s and 1970s America due to the growth of the politically correct scene. Since then, a shift has been made to use this term in a more social sense. Some natives, though, view it is another attempt to rename us again. As for the United States, I think a lot of us can agree that change doesn't always come easily here. Because Native American studies is not a widely popular field that schools teach accurately, you have some places that never move past seeing Indians as the stereotypes from the John Wayne movies. Other areas simply do not care. Despite the persistent use by the general American populace, though, Native Americans have grown to accept the term and continue to use it.

Simply put, the term "Indian" has a significant historical value due to its usage and many traditional Natives have become set on the term. Many Natives today have grown up with the term being used in many contexts and feel that it does not detract from their culture and/or individual identity. It all comes down to personal decision and many have decided to continue using the term. Whether it is "correct" or "inaccurate" is a subjective matter because, as pointed out above, the term "Indian" is technically correct under certain circumstances and is a legitimate term. Personally, I was raised with being called an Indian and my elders still use the term Indian, so I am fairly inclined to do the same. But the term I use can change, depending on context.

Edit: Added another sentence.

Names for peoples develop quite organically, you won't get very far trying to find a logical reason for it. However on this question I think today we find it especially confusing because we have much more of a sense of defined countries, and one of these is called India. It's a common misconception that early explorers (e.g. Columbus) thought they were in modern India. Columbus was aiming for 'Cathay' (China) and thought he had arrived at some islands somewhere near the coast. Yet he still referred to 'the Indies'. 'The Indies' he was referring to was a broad area of the Pacific off the east coast of Asia. Hence the names 'east Indies' and 'Indonesia'. See this map, from 1491, where this area is referred to as Oceanus Indicus Orientalis (Eastern Indian Ocean).

Clearly by the time the pilgrims landed in New England, people were aware that the east coast of America was not the 'east indies' either, but by this time the term 'indians' were quite well established terms the people living in the Americas, deriving from the initial Spanish colonisation. So this brings me back to my first point. Once something has a name, it's pretty unlikely it will be changed unless there's a reason to do so. It is a good question though why their name was not changed to 'Americans', as this would seem to be the best candidate.