Son of Manu is Priyavrata, son of priyavrata is Aagniidhra, son of Agniidhra is Naabhi, son of Naabhi is Rshabha, son of Rshabha is Bharata. The Nation is named after this Ruler par excellence Bharata known as Bhaaratam.
"अजनाभम् एतद् वर्षं भारतम् इति यत आरभ्य व्यपदिशन्ति" (भा.५-१ तः ७) This Nation known as Ajanaabha is henceforth called as Bharatam.
३) बिभर्ति लोकान् इति भरतः, भरतस्य इदं = भारतम् (One who protects the world is Bharata and that belongs to Bharata is Bhaaratam)
४) बिभर्ति स्वाङ्गम् इति भरतः, भरतस्य इदम् = भारतम् (One who protects his parts (world is treated as his part) is Bharata and that belongs to Bharata is Bhaaratam)
५) बिभर्ति = रक्षति, धारयति, पोषयति वा (In the above explanations the verb बिभर्ति means to preotect, to bear and to nurture/nourish
६) भरं तनोति इति भरतः, तस्य इदं = भारतम् (one who accomplishes (his) respomsibilities is Bharata and that belongs to Bharata is Bhaaratam.
Though this word is coined by the people or the followers of Bharta – the King par excellence son of Rshabha belongs to the race of Manu, probably this word Bharatam is coined by (seen literature of) Veda Vyaasa – the author of Puranas