What's the name Hakumen no Mono wanted to be called as?
My theory about its "real name" is, either Towako, the name of hakumen's avatar, maybe Hakumen named its avatar the real name for self-satisfaction? and the second probable name, is kirio/Inasa, as in Kirio Inasa, the boy whom it kidnapped. after the kidnapping, it probably change his name as it likes. 

I personally think, Towako is the "Real name".