The National Anthem is the first episode of the anthology show Black Mirror, where the Prime Minister is forced to have intercourse with a pig in ransom for Princess Susannah's return. But what does it have to do with the anthem? Why is it called "The National Anthem"?
There are two reasons to chose the title.
Firstly, It is a reference to the British National Anthem.
The British National Anthem is God Save the Queen. The episode's main plot is about rescuing princess Susannah. She is the next successor of the throne of the Great Britain.
Don't start now.
Whoever this is, is insane.
Prime Minister: And has a princess.
We'll get through this.
Wife: Not any princess - the princess.
The above are the words where The Prime Minister's wife says and emphasizes on THE. This means that the princess is important and the only one who is a successor of the throne. She would be the future queen. As the plot of the episode is rescuing her, the title is named such.
Another reason is that the incident has done something which an anthem would do that is make the people attentive and bring the people to a single opinion. People observe the incidents very curiously and attentively like they do to an anthem. So, this might be another reason why the episode is named such. [This seems less probable but not impossible].