Noticed a few patterns aside from the number thing in the Japanese version of the family tree.
Basically there's three types of names. 1. Names written in kanji with the character for a number
2. masculine names written in kanji
3. names written in katakana.
Seems like starting from the third gen they gave the first girl in each generation (aka the priestess) a number name, the boys regular* kanji names, and every girl after the first one a name written in katakana (not sure of the connotations there, but maybe it's to note that they're supposed to support the priestess.)
* The founder of the Hirumori clan, Sutekichi, has a weird name. The first character(捨) means discard, throw away, abandon, sacrifice, etc. I think there is a thing where you give a kid a name like that so demons won't want them, but considering that male children are adopted into the Hirumori clan maybe it's just a mean name.