Once created a respository named: name.github.io then added a CNAME would I be able to rename the repository without it breaking
If your DNS are pointed to github and your CNAME file is ok, it will not break. You will just have to wait for the changes to replicate in Github architecture.
But, you will be changing from a user/organization site to a project site. That mean that your site will have to be in a gh-pages
branch.
You also will have to add an empty .nojekyll
file at root to instruct Github pages not to process your site as a Jekyll site. This is important if you have .
or _
prefixed files or folders.