How do you fork your own repository on GitHub?

A super easy way to do it in 30 seconds from the GitHub website:

1. Copy your repo's URL. Ex: https://github.com/YourName/YourOldRepo (hint: it's the URL when you look at your repo's main page on github.
2. Click the + icon in the top right corner.
   
3. Select "Import repository".
4. Where it asks for the "Old URL", paste the URL you copied at step #1
   
5. Enter the name of your new repo and click Begin Import.
6. That's it! You now have a copy of the full repo, with all commit history and branches!

Limitations: It's not actually a real fork. It's a copy of the repo. It won't allow to do pull requests back and forth.

Simplest way to achieve the desired effect is to create a new repository, then select the import option and supply the URL of the repo you wish to fork.

Images below will help:

Fork own repo via import-1

Fork own repo via import-2

1. git clone https://github.com/YOURREPO.git TargetRepoName
2. cd TargetRepoName/
3. git remote set-url origin https://github.com/TargetRepoName.git
4. git push -u origin master

I don't think you can fork your own repo.
Clone it and push it to a new repo is good but you need to:

git clone https://github.com/userName/Repo New_Repo
cd New_Repo
git remote set-url origin https://github.com/userName/New_Repo
git remote add upstream https://github.com/userName/Repo
git push origin master
git push --all

(see git push)

See the all process described at "Fork your own project on GitHub".

---

Six years later (2016), you now have the GitHub importer which allows you to import a repo from another source... including GitHub.
See "Importing a repository with GitHub Importer"

narf's answer (upvoted) also illustrate that process.

That will allow you to create a new repository and import the full history of the old one into the new one, using its GitHub url.

Again: what you get is a copy, not a real fork: you cannot make pull request from the new repo to the old one.

Again (bis), as stated in the comments by mpersico, this is not a TRUE FORK.

If I have a foo which is the canonical source repo for an open source project that I want other people to fork and have access to do PR, then I do not want to work in that repo, I want a fork I can use to issue proper PRs against my project.
I have solved this my creating a second account in GitHub and forking to that.