Copy/fork a git repo on github into same organization
If you do not need the fork relation (e.g. you want some kind of decoupled alternate repo for whatever reason), duplicating the repo as outlines by your Google finds and larsks's answer is fine.
If you do want to make it a fork, contact Github support ([email protected] or https://github.com/support), and they will create a fork in the same organization for you. (They're not picky about this either, you'll have just to provide an alternative name for the repo, as repo names within an account must be unique.)
Update: User Steve Rice reports in the comments below that GitHub Support stated that support would not currently/no longer set up a second fork in your account.
GitHub recently posted an article about possible alternatives to forking a repo into the same account. You can read that article here.
Just create a new repository and push to it from your working copy:
git clone [email protected]:me/myrepo-original
cd myrepo-original
git remote set-url origin [email protected]:me/myrepo-new
git push origin master
Now you have a new repository, myrepo-new, which is identical to myrepo-original.