I think it may be possible that the indian founding fathers Nehru, Gandhi etc.. were acutely aware of the lack of identity that the people of hindustan had or maybe they just got lucky and inherited the continuing legacy of what Colonial Britain left in their ''creation'' of the country india.
For all intent and purpose, india is an artificial country in a modern context. It never existed historically, nor does it have any base. Many biased indians will point to Ashoka's empire, but it was not referred to india either, not on any maps, not by Ashoka himself. No ancient map exists ever showing a place called ''india'' in this form, infact alternate names are used to describe the various parts of South Asia.
As can be seen all throughout history is that the people inhabiting the gangetic plains and the subcontinent peninsula down to between the eastern and western knats have been very astute imitators of foreign cultures.
The people have always had an inheritent complex of identity, and were quick to adopt foreign cultural practises and traits, over time, gradually claiming those norms for themselves erroneously. This trait is applied over and over again all throughout history and may explain the inherent complex that many people have in South Asia in general, and as can be seen in this forum in general, the large numbers of indians who are Pakistan-fixated or fixated on other foreign things not indigenous to them.
Look at ancient hinduism (Note: the term hindu isnt even indigenous and was never used by the ancient practisers of ''hinduism'', it was introduced by foreigners after the 16th century - this is another discussion in itself)
The religion was introduced and shares many similarities to the pagan Iranic (Aryan Tribes) the settled in Pakistan and Afghanistan. In fact, Aryan legend holds that they settled the trans-indus and later Panjab region/eastern Afghanistan from where there religion spread in various directions leaving an impression. what is interesting is that many scholars, notably Iranian ones but also others, note that while hinduism shows many similarities and most likely is derived from the ancient Iranic faiths, the people now practising it are definately not Iranic in the genetic sense. In essence, the inhabitants of hindustan have adopted a foreign religion and given it a local flavour(=hinduism) claiming it as their own.
Many foreign traditions, introduced by Central Asian turks, Afghans, Arabs in the form of music, dance, art, culture, mannerism have left their imprint on them. But ironically, up until even 1526 when Babur defeated the Delhi Sultanate (Battle of Panipat), he noted that the people of Hindustan lacked basic social skills and rules of engagement. For a central asian turk to speak of such findings, doesnt point to any ''civilization'' at all. Ironically, go to wikipedia and you'll find indians editing it to read, that Babur was ''an indian ruler''. A better read would be Baburnama which gives considerable insight into the mindset of the native aboriginal people and his impression of them.
For the Hindustani, the best thing imaginable, was the arrival of the British, who not only put an end to the longstanding rule of the Mughals (A central asian empire) but also of several other empires (Hyderabad's Asif jahs), Nawabs of Bengal, Afghan etc... but built up the continent with a series of massive infrastructure projects, communication and railworks, schools and training, empowered them. They applied the name india, previously used for a region straddling the 2 banks of the Indus river in Sindh in ancient Greek and Persian maps, and applied to the entire continent stretching from burma to eastern Afghanistan.
In 1947, left them with a country. In turn giving the people of Hindustan a foreign name, new found identity and new found culture, language(Hindi over indigenous languages; note: interestingly, Panini, the ancient sankrit master was from Ghandara, Pakistan)) a sense of history and civlization, tying them back incorrectly to the ancient times. Winston Churchill summed it up just right when he said that "India is a geographical term. It is no more a united nation than the Equator". Despite this, people living all the way in Bengal, Andra Pradesh, UP and Tamil Nadu, somehow feel ''connected'' with people from the indus river of Pakistan?? sounds absurd but not when you have generations of absorbing foreign cultures and claiming them as your own. It is said, that those without any history or culture are quick to point them out on other people and find remote links to themselves so as to improve their stature and origins... Sounds familiar (I'll just leave that one as that)
Wether ''indians'' themselves actively picked the name india or were meagre bystanders in the colonial game of cultural hijack and manipulation, distorting the history and facts about the region, one will never know for sure, but what can be said is, judging by the number of people who continue to discount this and are vehemently in opposition to it, the british did a good job at it!
By the way, this is an interesting topic, I hope no one is offended by it, as it has considerable academic merit and people need to discuss it and be more aware of this cultural hijack and sabotage (one of many) inflicted by the British on the local peoples of South Asia