Making a private github's wiki public

Fork the repository, delete all contents by creating a new orphan master branch, force-pushing master and pruning all other branches, make the fork public.

Alternatively, create a new empty repository, create a dummy wiki page on that new repostitory, check out the wiki of the old repository and push it to the new:

git clone [email protected]:user/old-repo.wiki.git
cd old-repo
git remote add new [email protected]:user/new-repo.wiki.git
git push -f master new

Both these solutions result in two repositories: The old, private one that contains the data, and the new public one with the wiki. You can then delete the old wiki.