Jambudeep: Jambudvipa was used in ancient scriptures for the name of India before Bharata became the official name scriptures began using.
Bharatavarsha or Bharat: According to the Puranas, this country is known as Bharatavarsha after the King Bharata. This has been mentioned in Vishnu Purana and many other puranas along with brief in Mahabharata.The term varsa means a division of the earth, or a continent. The country (varṣam) that lies north of the ocean and south of the snowy mountains is called Bharatam;
Aryavarta: Aryavarta means the rule/land of Aryans. In the map Aryavarta is a land between Himalaya and Vindhya ranges.
Hindustan: Commonly used at the time of Mugals which was given by Persians. Persians called India as 'Hindustan' or the land of Hindus. Hindu was derived from Sindhu as Indian civilization was on banks of Sindhu.
India: Ancient Greeks called Hindustan India which is current name in English. India was called such because Indian civilization was near Indus.
Some less common names in other civilizations
Hodu: Hodu is the Biblical Hebrew name for India mentioned in the Book of Esther.
Tianzhu or Tenjiku: Tianzhu or Tenjiku is the historical East Asian name for India that comes from the Chinese transliteration of the Persian 'Hindu'.
Al-Hind - Used by Arabs for the Indian country which were assumed to be hindus.
Meluha - Used by Sumerians