Analysis of information sources in references of the Wikipedia article "List of people known as the Great" in English language version.
After the coronation of the child the two Leos would be distinguished as Λέων ὁ Μέγας and Λέων ὁ Μικρός, and this I believe, must be the origin of the designation of Leo as "the Great"; just as reversely Theodosius II. was called "the Small," because in his infancy he had been known as ὁ μικρός βασιλεύς to distinguish him from Arcadius. Leo never did anything which could conceivably earn him the title of Great in the sense in which it was bestowed by posterity on Alexander or Constantine.