Ἡγουμενία according to H.Stephanus is "dignitas abatis", the rank of an abbot [ἡγουμένος ], the abbacy. The term, although used exclusively in the Byzantine age, is to be found in Aelius Herodianus, who likewise advocates its spelling with diphthong (i.e. rather than ἡγουμενία ).
The entry is taken from Lexicon Ambrosianum (Ambrosianus 37, but not in Laurentianus 59,16). Perhaps originally taken from Aelius Herodianus Partitiones 45 or 221.