I have only a small contribution to make. There doesn't seem to be any Greek goddess named Ara
According to this, there are demons in Greek mythology called arai
). The word apparently means "curses", and is a group title, not a name.
The name for the parrot or macaw seems to come from a Tupi (native South American) word. In any case it is irrelevant since people aren't normally named after types of macaw.
As for Ara
's use as a feminine name in the USA, if you check the popularity data, Clara
and Sarah/Sara were very popular names at that same time. Cara
were also in the Top 1000 in the same time frame. Perhaps Ara
's usage at that time could be explained by its trendy sound. Maybe Ara
could have been a shortening of Arabella
(both were in the Top 1000 at the same time as Ara
), or have come directly from the Latin word ara