At the times of the Mishna and Gemara, one of the more commonly used names to refer to God among Jews was RaḤmana, literally the Merciful. It seems like this was widely used as Hashem is today. When and why did usage of this name fall out of popular usage?
I would vote for this explanation. Calling God 'Rahmana' comes too close to the invocation Muslims recite all the time, especially before reading a chapter of the Qur'an: بِسْمِ ٱللَّٰهِ ٱلرَّحْمَٰنِ ٱلرَّحِيم, -- Bismellah el Rahman el Rahim, meaning "In the name of God, the Gracious, the Merciful." So when Islam emerged in the 7th century, Jews slowly stopped using it.
I have no proof of this, just as I have no proof that giving a boy the name Yishmael (note: the great tanna Rabbi Yishmael) fell from favor and disappeared in Judaism after the rise of Islam.