As an example: suppose you have a personal project like Angular Seed (which is a starting point for an Angular project https://github.com/angular/angular-seed).
Now
I just found this answer https://stackoverflow.com/a/4096529/131227 as well as a comment (git workflow - using one repo as the basis for another) which together lead me to a solution I like.
git clone -o boilerplate ssh://git@github.com/user/proj.git new_proj
The -o boilerplate renames the origin to boilerplate
which can still be used to pull changes from.
Create your new empty github new_proj repo.
Then
git remote add origin ssh://git@github.com/user/new_proj.git