Well I know that 'Hindustan' was called 'Bharat' after the name of king Bharat, but curious to know what was the ancient name prior to that?
The name was Ajanābha-varṣa. This was the name given to the planet earth. But after the activities of the king Bharata, the country is named as Bhārata-varṣa.
This is said in the Śrīmad Bhāgavatam. Skanda 5 chapter 7
ajanābhaṁ nāmaitad varṣaṁ bhāratam iti yata ārabhya vyapadiśanti.
Formerly this planet was known as Ajanābha-varṣa, but since Mahārāja Bharata’s reign it has become known as Bhārata-varṣa.
This country was named after King Nābhi. He is the grandfather of King Bharata. After the reign of Bharata, son of Ṛṣabhadeva, it is being called as Bhārata-varṣa.