Hi Paul,
...what are the advantages of GIT I can point to over other version control systems such as Subversion?
I'm glad a few others chimed in. For this question, I generally refer people to this very old post on the main Git wiki:
https://git.wiki.kernel.org/index.php/GitSvnComparison
The two key aspects mentioned so far are ones I regularly offer: 1) distributed nature (i.e. local history/commits without the need for a server), and 2) fast branching.
There are ways for an enterprise to host a centralized Git hosting solution (GitHub's Enterprise product, GitLab), and this might offer a path towards using it. Some enterprises may not be aware that they can host Git entirely within their network, and this may be a good first step for a trial.