The inhabitants of Hinamizawa believe that the blood of demons flows in their blood. As such the character for 'demon,' 鬼, is treated as very sacred.

For example, usage of the character 鬼 in names is apparently a right reserved for only the leaders of the Kimiyoshi and Sonozaki families.

Case in point, we can see it in the current Sonozaki family leader's name—Oryou—is written with the character 魎.

This, and only this, is proof that she is a legitimate leader of the Sonozaki family.

In the same way, Mion Sonozaki, its successor,

contains 鬼 in her name, with 'Mi' being written as 魅.

It also goes for Oryou's daughter and Mion's mother, Akane, who did not succeed her.

It's known that before the disowning, Akane's name was actually written as 蒐, though it was changed afterwards to 茜 (still read as 'Akane').

Incidentally, it's thought that the first character the mayor Kiichirou's name, 喜, originally meant 'demon,' which can also be read as 'ki.'

It should be said that names using the character 鬼 aren't limited only to successors of the Sonozaki family.

You can even find it in the last names of the Three Families.

For example, the characters for 'Kimiyoshi,' 公由, are likely a split-up version of 鬼. If you write it in reverse and top to bottom, as 由公, it clearly resembles the character 鬼.

As for the Furude family (古手) who have been in the Shinto priesthood for generations—the word for 'fortune-teller' can be written, without interval characters, as 占手.

I think that with the addition of a demon's horn—the tick on the top of the character 鬼—the 占 became the 古 that is used in their name now.

In the case of the Sonozaki family, where Sonozaki is written 園崎, I can't find any reference to the character 鬼, perhaps because it's already ingrained into their first names.

They were the clan responsible for running the Watanagashi ceremony, so I think that instead, they included the contents of that ceremony into their last name.

The 'zaki' in 'Sonozaki' is a conjugation of the word for 'slice,' and the 'sono,' written as 園, probably contains the hidden meaning of 'human body,' as it is a complicated mess (organs) enclosed by four sides.

I think, in other words, that 'Sonozaki' comes from 'those who slice open the gut.'