TL;DR: I don't think we should change it. One of the main purposes of our site is to provide authentic information about Hinduism and most people only know Vedic Religion by the name Hinduism.
As the question asks:
Before Down-voting or Criticizing please express your views.
I'm expressing my views here.
First understand the term:
Sanatana means Eternal and Dharma means Religion.
Why Hinduism/ Vedic Religion is Sanatana Dharma ?
There are mainly two reasons for calling Hinduism/ Vedic religion as Sanatana Dharma:
1) It has no origin. It is coming from time beginning. And it was even in previous creation. RigVeda 10.190.3 states "Ordainer created Sun and Moon like that of previous creation."
2) It is based on Vedas and Vedas are recompiled in every Dwapara by Veda Vyasa. As one can see in answer here there have been 28 Veda Vyasa only in this Manvantara. There were 1131 Shakhas of Vedas and upto now all are lost and we have only about 10 Shakhas of Vedas. We have lost various texts of Samhitas itself. But even with this loss Vedic text will not be lost, because Ashwatthama is going to become next Vyasa in next Dwapara and he will recompile all those lost texts. So as in every Yuga a person comes as a Vedavyas and restores lost text of Vedas and revives Vedas, Vedas are not lost.
As Vedas are not lost due to this cyclic nature of their revival of the same texts. So, it means religion based on Vedas will also remain eternal.
In this way Vedic religion is eternal and it comes under category of Sanatana Dharma ie. Our religion Vedic Sanatana Dharma is Eternal.
Is Hinduism/ Vedic religion the only Sanatana Dharma?
As we discussed in above how Vedic religion is eternal. We discussed two main points there:
i) It should not have any origin point.
ii) It should be revived as cyclic phenomena, so that religion is never lost.
So, Is Vedic Religion the only religion to fulfill this Criteria?
If we see only with respect to Vedic scripture we can say yes. But if we judge with religions of their scripture. We come to the answer No.
Most people might think that Siddhartha Gautama who latter became Gautam Buddha is the founder of Buddhism. And as it was founded some 2500 yrs. ago and as we see it's origin it isn't eternal and thus Buddhism is not eternal religion. But Is this true with respect to Buddhist scripture?
A Big No.
Buddhism doesn't regard Gautam Buddha as founder of Buddhism. But he is just one among the many Buddhas.
There were previous Buddhas before the present Buddha.
Actually as per Buddhist texts Present Buddha ie. Gautam Buddha is 4th Buddha of this Kalpa. The 3rd Buddha being Kassapa Buddha , 2nd Buddha being Konagamana Buddha and the 1st Buddha of this kalpa being Kakusandha Buddha. And the upcoming ie. 5th Buddha of this Kalpa being Maitreya Buddha.
Present Buddha Maitreya is living in Tusita heaven now and will come to earth when teachings of present Buddha is lost and again revive Buddhism.
Just as Hinduism calls this Kalpa as Shweta Varaha kalpa, Buddhism calls this Kalpa as Bhadra Kalpa and it is also of duration of billions of years. Buddhist text also provide a list of 27 previous Buddhas including Buddha of Previous Kalpas also.
Thus Buddhism also has no origin and it was also in previous creation and it is also revived again and again by coming Buddhas from Tusita loka (as per Buddhist text). Thus Buddhism is also Bauddha Sanatana Dharma.
Most people again think Mahavira Jain is the founder of Jainism and as it has origin it can't be called Sanatana Dharma. Again this is wrong as per Jain scripture.
In Jainisim also Mahavira Jain is only one among the 24 Tirthankars of Jain. ie. He only Revived Jain Religion.
Just as Veda Vyas comes in cycles in Hinduism, similarly in Jainism 24 Tirthankars come in one cosmic cycle of this world. The present or 24th Tirthankara being Mahavira Jain.
Actually we have also some historical records of 23rd Tirthankara named Parshavanatha
So, Jains strongly believe in cyclic revival of Dharma. They also have concepts of Vasudeva, Baladeva and Prativasudeva.
So, Jainism also fulfills the criteria of being called Jaina Sanatana Dharma.
Although Quran claims that there were messengers sent by Allah before Prophet Muhammad all over the world. But Quranic text declares that Prophet Muhammad is the last messengers of Allah. Thus it stops the possibility of cyclic revival of Dharma. Neither there is a statement that there were Previous Muhammad who gave Quran. Nor a person named Prophet Muhammed will come to give Quran again as Prophet Muhammad is considered the final messenger.
So, it doesn't fulfill our criteria of calling Islamic Sanatana Dharma.
Although there is a concept in Christianity about 2nd coming of Jesus. But nowhere it is told that there were previous Jesuses and they come like in cycle to revive Christianity again and again. Also we do not find statements of cyclic nature of creation and Jesus giving Bible in previous creation.
So, it doesn't fulfill our criteria of calling Christian Sanatana Dharma.
So if we go by scripture of respective religion, we find Vedic Religion, Bauddha Religion and Jaina Religion as Sanatana Dharma. Both Jainism and Buddhism also uses the Sanskrit words and the terms Sanatana and Dharma. Buddhism also uses the pali term Dhamma to denote it. Thus we have three religions in the list of Sanatana Dharma:
1) Vedic Sanatana Dharma.
2) Bauddha Sanatana Dharma.
3) Jaina Sanatana Dharma.
So, the precise term for our religion becomes Vedic Sanatana Dharma and not only Sanatana Dharma.
There is no proof that there were previous Buddhas and previous Tirthankaras (except Parshavanatha). So it is not true.
There is also no proof that there were previous Veda Vyasa. We are telling it as per our scripture and they are telling it as per their scripture.
Why should we believe in their scripture?. We believe in Vedas which are eternal.
Why should they believe in our scripture? They believe in their scripture. And they do not believe Vedas are eternal. Why should they believe?
So we should change our site name to Vedic Sanatana Dharma.?
I don't think we should change it. One of the main purpose of our site is to provide authentic information about Hinduism and most people only know Vedic Religion by the name Hinduism.
In such condition if we change the name of Hinduism to other site it will become less popular. And most people will be trapped in many degraded answers which are present in Quora and other site. As they use the name Hinduism and people will be misguided.
As one of our main purpose is to provide authentic information about Hinduism to the world. Hence I believe we should act in that way. Hence I believe we should keep our site name as same. ie.
This is just only my View. People may agree or disagree. And the community members may decide further in this matter.
One may comment whether one agrees or disagrees with my Views.