Sesshoumaru's Mother's Name
Sesshoumaru’s Mother has no official name.
She has gone by “Gobodou-sama” Which is A formal way to address one’s mother or in the English Dub as "My Lady". It can also translate to something such as "Lady Mother"
I have seen fans call her "Irasue", which is a made up fan name. We already have a character named "Urasue".
The most popular fan made name is "InuKimi"
We know that "Inu" means "Dog"
For "Kimi" there are different meanings: A sensation; You, buddy, pal; Egg yolk; Millet, Monarch, Mister, Ruler, or the kanji can be used also as a Male Name Suffix "-kun"
Though we wouldn't know until we found the actual characters used for this fan made name.
My friend Kari, who lives in Japan, and speaks Japanese said that
"君 is also used relating to monarchs as well. Japan's national anthem is called "Kimi ga Yo"
She was able to tell me the actual origins of the fan name "InuKimi":
She said “I had heard the name was 犬君 but overseas fans misread the characters and pronounced it "Inukimi” but actually the pronunciation is “Inuki”
“Inuki" is a young girl, who serves Murasaki, one of the female leading characters in The Tale of Genji (in the Heian Period) (794-1192). In the Tale the girl is so full of curiosity that she sets free a caged bird and makes Murasaki feel so sad.
This is the character that Sesshoumaru's Mother's Fan Name is inspired by.”
I asked Ayuuria why it is pronounced as InuKi, and she told me the reading is in the parenthesis 犬君(いぬき)
So when you seperate the Kanji it will read as 犬 君 Inu kimi
Together it will read as 犬君 Inu-kun
With a masculine honorific
But the origins of the fan name is inspired by the Tale of Genji by Muraski Shikibu which is pronounced Inuki 犬君(いぬき)
This character Inuki appears in The Tale of Genji
The Tale of Genji: Volume 2 p.70
The Tale of Genji: Volume 3 p.216-217
The Tale of Genji: Volume 4 p.178, 194
The Tale of Genji: Volume 5 p.21
The Tale of Genji: Volume 6 p.137
The Tale of Genji: Volume 7 p.95
https://uenashi.com/the-tale-of-genji/?fbclid=IwAR04lpMZgWGmFOW9aNyZmIeanbrHdXtF1wUd1ejP4GhGk8ANmaTgrsNvJiA#:~:text=%E3%80%8C%E8%8B%A5%E7%B4%AB%E3%80%8D%E3%81%AE%E5%B7%BB%E3%81%AB%E3%81%8A%E3%81%91%E3%82%8B%E7%8A%AC%E5%90%9B,-%E7%B4%AB%E3%81%AE%E4%B8%8A&text=%E3%81%93%E3%82%8C%E3%81%AF%E5%85%89%E6%BA%90%E6%B0%8F%E3%81%8C%E3%81%BE%E3%81%A0,%E3%82%92%E9%80%83%E3%81%8C%E3%81%97%E3%81%A6%E3%81%97%E3%81%BE%E3%81%84%E3%81%BE%E3%81%99%E3%80%82
https://jisho.org/search/%23kanji%20%E5%90%9B
There you have it. That is the origins of the fan name. Although, Sesshoumaru's Mother has no Official Name as of now.