The call to refer to the program as the government option instead of the public option came from Republican pollster Frank Luntz, Media Matters and Kurtz report. But this shouldn't disqualify the new term from the Fox News stylebook. Government option is superior to public option in that it emphasizes that the government—and thus the taxpayers—will be footing the bill. As a modifier, public has many nongovernmental uses, as in public appearance, public figure, public display, public-key cryptography, public editor, public enemy, public storage, and public opinion.
But when government is used as an adjective, there is no such confusion.