The commonly accepted idea is that we were given this term by outsiders, and it means 'people east of the Indus' or something like that.
Is it important?
I get the impression that for nationalists the idea that outsiders gave this name is unacceptable. That it somehow diminishes Hinduism. It has become a matter of national pride, to distance oneself from things foreign.
I think these feelings should be taken seriously. Then again, it is often the case that our names come from others. I did not invent my names, they were given to me. Yes there are people who invent their own names, mostly artists, and that is a sign of vanity, not greatness.
If you look at the original native names for ancient peoples and cities, in their own language they often mean "the people" and "the city" or "the big city". This word is either mispronounced by foreigners or they use a description of the people or city in their own language.
Like The English name "Egypt" is derived from the Ancient Greek "Aígyptos". The Greek forms were borrowed from Late Egyptian (Amarna) Hikuptah or "Memphis", a corruption of the earlier Egyptian name, meaning "home of the ka (soul) of Ptah", the name of a temple to the god Ptah at Memphis.
The ancient Egyptian name of the country was km.t, which means black land, likely referring to the fertile black soils of the Nile flood plains. This name is commonly vocalised as Kemet, but was probably pronounced [kuːmat] in ancient Egyptian.
Understandably, after decolonization it became something of national pride for many peoples to restore names in their own language again. For instance Chinese no longer accepted Peking, but want it to be called Beijing. Wikipedia says this
The name Beijing, which means "Northern Capital", was applied to the city in 1403 during the Ming dynasty to distinguish the city from Nanjing (the "Southern Capital")
Now for Hindu Nationalists I guess the best explanation would be that Hindu is a name given after the ancient King Indu, the ruler of a great Kingdom, who lived 10.000 years ago, and founder of the first civilization. Alas, we are not always so lucky.
It is a sign of a nation rising. In the old days people rising felt the same national pride. For instance the Romans succeeded the Greek and adopted much of their culture from them. But besides admiration this left them with an inferiority complex and fierce jealousy. So when they had become a big empire the emperor appointed their best poet Virgil to write the history of the Romans. So he created the exodus story of Romans being the descendants of a Prince of Troy, the Trojans being the famous arch-enemies of Greek. So the Romans could say: The Greek may have beaten the Trojans, but in the end we Trojans were victorious over the Greek.
I often wonder if the exodus story of the Jews is not a similar writing to give the Jews a noble past. I find it remarkable that the oldest known writings are in Greek and created in Greece, where learned Jews picked up sophisticated writing. Jews were telling the Greek they were the builders of the great pyramid. Of course they could not know that this would be debunked later on. A little exaggeration was not strange to our ancestors. There is a natural tendency to paint a great past when people rise to greatness. In the same way that people from humble descent that get rich, change their names to suggest more noble descent.
In the same way the later European colonial elite felt the need for a great past to fit their empire. So the created this myth of being the Aryans who had brought civilization to ancient India, regarding themselves as their descendants who were still bringing civilization all over the world. This idea was popular all over Europe.
Indians can be very angry about this western appropriation, but they can also interpret it as a recognition that it is the Indians that have the older civilization. If there is any people on this planet that do not need to touch up history, it is the heirs to the eldest uninterrupted civilization. So to me this all feels a bit unnecessary.
