Or did they use another title to claim themselves rightful successor to him?
Just to clarify, Chinggis is a name, not a title. Khan is a title.
His descendants did inherit his title, that of Khan: Batu Khan, Jochi Khan, etc.
Its an Asian (Turko-Mongolic-Indo-Persian) tradition that rulers used to chose a name/title at his coronation by which he chooses to be known and remembered other than his original or given name. Let me give you examples of Mughals for this, the first mughal Emperor Babur was named‎ Zahir al-Din Muhammad but chose the name Babur (means Tiger in Persian) at his coronation. Similarly Emperor Akbar was named Abu'l-Fath Jalal ud-din Muhammad but chose Akbar (meaning Great) as his name upon coronation. Much like popes.