The original name for the series in Japan is Ninja Ryukenden (translated as "Legend of the Ninja Dragon Sword"). When translating the game for an English audience, the team had a hard time finding a way to translate the title. In the end, they decided to rename it Ninja Gaiden "because it sounded cool."
Despite the North American name of the series being Ninja Gaiden (gaiden being the Japanese word for "Side-Story"), the series was not originally a spinoff or part of another series. This (arguably) changed when Ryu Hayabusa was added to the Dead or Alive canon, as well as having Ayane appear in the newer Ninja Gaiden games, making both series' stories intertwined.