The Vespucci expedition also named the Potengi river , `` Rio Grande '' . The name is Portuguese for `` Great River '' . The Vespucci expedition also named the Potengi ( Tupi-Guarani for `` River of Shrimps '' ) river , whose considerably large mouth contrasted with the nearby bodies of water , `` Rio Grande '' ( Portuguese for `` Great River '' ) , after which the Captaincy , Province , and State were named .