If you own a gist, for example gist number 123456, then you can clone it locally using the scp-like git url syntax and push up your changes to the remote gist.github.com.
# clone your gist
git clone firstname.lastname@example.org:123456.git my-awesome-gist
# edit some file, commit changes
# e.g. git commit -v some-file.txt
# then push it back up to gist.github.com
# for everyone else to access