It is already 20th and I haven’t written about different names for October yet!
Usually it is 十月 (じゅうがつ / juugatsu) but the old name is 神無月 (かんなづき / kannaduki) – the meaning of each kanji is: 神(gods)_無(no)_月(month).
There are many theories why it’s called that way but they can be divided into 2 major groups. One is that this “no” is the old way of writing the particle の (no) in the same as that of 水無月 (minaduki / June), so 神無月 means “the month of gods.”
The other group’s theories are something like “all gods go to Izumo Shrine (Shimane Prefecture) for an annual conference and they are no gods anywhere else.” In fact, in Izumo, October is called 神有月 (kamiariduki) and they have a series of festivals.
Shimane Prefecture is at the western end of Honshu, the main island, and Izumo Shrine is one of the oldest Shinto shrines. If you have a chance, go and visit it!