SharePoint Online - name change of a sub-website

Silas_Muth 

IF by Site Name you meant Site Title,

then Site Title is multi-lingual in SharePoint, in case if multiple languages are enabled in the site then you need to change Site Name / Site Title for all the language.

Go the the Site Settings  Language Settings there you can see enabled language for the site.

Below are the steps which you need to perform in order to change Site Title for each language.

Manually

• Take one user and set your language (for which you want to change Site Title) in user profile property language and set priority accordingly.
  • Go to Office Profile from People Menu

• Click on Update Profile

• • Click on language and regional settings

• • Go the language and region and set your language

• Navigate to your site and now you will see Site in the language which you have set as first priority in User Profile (This might take 10-15 minutes to reflect -> and try to re-opening browser)
• Now change the Site Title of the site
• Repeat above steps for all the available languages 

Backend (CSOM)

Reference Link: 

https://docs.microsoft.com/en-us/sharepoint/dev/solution-guidance/create-sharepoint-content-types-by-using-csom#localize-content-types-lists-and-site-titles 

Hope it will helpful to you.