When a multibranch job is configured with a GitHub branch source the repository field contains only public repositories of the selected owner.

The current workaround which works for me is modifying the select field (using FireBug) by dynamically changing an option to the required value (that is, the name of the desired private repository). After submitting the form this private repository is correctly used.

Of course it would be much more convenient if also private repositories would be contained in the select list.