Sanatana Dharma originated in India. India also has another name: "Bharata/Bharathavarsha". The term ArshaBharata culture is commonly used for pointing out the Sanatana Dharma in South India. How did the nation get this name?
After the name of an ancient king Bharata, India is also known as Bharata-varṣa since the prehistoric times. Before it, this land was called as Ajanābha-varṣa. This is stated in Shrimad Bhagavatam:
ajanābhaṁ nāmaitad varṣaṁ bhāratam iti yata ārabhya vyapadiśanti. [SB - 5.7.3]
The name of this land was Ajanabha. It came to be known as Bharata after starting of his (King Bharata) reign.
Besides Srimada Bhagvatam, other texts also mention the same reasons. Just adding here for references.
भरतोनाम राजाऽभूदाग्नीध्रः प्रथितः क्षितौ ।
यस्येदं भारतं वर्षं नाम्ना लोकेषु गीयते ॥ २ ॥
- There was a great king named Bharata on the earth. He was the son of Agnīdhra. This Bhāratavarṣa is named after him.
Chapter 14, Anuṣaṅga-pāda, Brahmāṇḍapurāṇa
सोभिषिच्यर्षभः पुत्रं महाप्रव्रज्यया स्थितः ।
हिमाह्वं दक्षिणं वर्षं भरताय न्यवेदयत् ॥ १४.६१ ॥
तस्मात्तु भारतं वर्षं तस्य नाम्ना विदुर्बुधाः ॥ १४.६२. क ॥
- 61,62a. Ṛṣabha crowned his son and engaged himself in Mahāpravrajyā (the great migration of renunciation i.e. journey or pilgrimage till death). He allotted the southern sub-continent named Hima to Bharata. Hence learned men know this sub-continent as Bhāratavarṣa after his name.
As per this text translation by G.V. Tagare, the previous name of the Bhāratavarṣa was Hima.
Apart from the King Bharata the son of Rishabha mentioned in the other answers there is an alternate theory also.
Since the country (Varsa) is called Bharata the people are Bharatis. Since Manu  ruled over the people, he is called Bharata. Etymologically derived thus the Varsa (country) is called Bharata.
Vayu Purana I.45.76
 This etymological explanation of ‘Bharata’ attributing its credit to Manu who is called ‘Bharata’ due to his act of maintaining (and feeding the people) supersedes the Jaina tradition which attributes it to Bharata, the son of Tirthankara Rishabha