Among the Mongols themselves and various states in East Asia, the Mongol empire was called the "Great Mongol State," but only after Chinggis Khan's death. In Mongolian, this was yeke ulus ("great state"); in Chinese, daguo 大國; in Jurchen and Manchu, amban gurun. As for where this title came from, Timothy Brook has an article dedicated to it, which I've cited below.

There are two origins for the practice of calling a large-scale state "great" in East Asia/Mongolia. The first is from Chinese naming practices. The Song Dynasty (960-1279), which preceded the Mongol Yuan Dynasty (1279-1368), sometimes referred to itself as the "Great Song State" da Song guo 大宋國. The Jin and Xia, both states on the northern frontier in competition with the Song, also called themselves "great". There are earlier references to "Great Tang" (618-907) and "Great Han" (206 BCE - 220 CE), but these are fairly scattered to my knowledge.

The second origin comes from the Mongols themselves. At the Great Assembly (or kurultai) of 1206, Chinggis Khan asserted that he ruled not simply the Mongol ulus (Mongol state), but a greater ulus "under the protection of Eternal Heaven". This greater ulus included all "people of the felt-walled tents" (nomads) and "people of the wooden doors" (sedentary agriculturalists), and is related to the Mongol concept of "empire" which claimed everyone in the world was either part of their empire, or not-yet-part-but-would-be-soon. No text from the time specifically refers to Chinggis Khan's empire as the yeke Mongol ulus (Great Mongol State), but the nomenclature was appended by his successors and the enemies of the Mongol empire.

Sources:
Timothy Brook, "Great States," in The Journal of Asian Studies 75.4, November 2016, pp. 957-972

EDIT: It would be grand if someone with a background in European or Islamic history could add their perspective and knowledge.