If you are the only person working on the project, it's not a big problem, because you only have to do #2.
Let's say your username is someuser and your project is called someproject.
Then your project's URL will be1
git@github.com:someuser/someproject.git
If you rename your project, it will change the someproject part of the URL, e.g.
git@github.com:someuser/newprojectname.git
(see footnote if your URL does not look like this).
Your working copy of Git uses this URL when you do a push or pull.
So after you rename your project, you will have to tell your working copy the new URL.
You can do that in two steps:
Firstly, cd to your local Git directory, and find out what remote name(s) refer to that URL:
$ git remote -v
origin git@github.com:someuser/someproject.git
Then, set the new URL
$ git remote set-url origin git@github.com:someuser/newprojectname.git
Or in older versions of Git, you might need:
$ git remote rm origin
$ git remote add origin git@github.com:someuser/newprojectname.git
(origin is the most common remote name, but it might be called something else.)
But if there are lots of people who are working on your project, they will all need to do the above steps, and maybe you don't even know how to contact them all to tell them. That's what #1 is about.
Further reading:
Footnotes:
1 The exact format of your URL depends on which protocol you are using, e.g.
