Why is Miller's case titled R (Miller)?
R (Miller) v Secretary of State for Exiting the European Union
The meaning of R is Regina (Queen). This is confusing me. Wasn't the case brought by 'an individual(s) v the government'? Why is the title 'government (individual) v government'?
The Queen (or King) is not the government; she represents the State. The difference is often ignored by ministers, but is important particularly in constitutional cases.
Miller began as Miller v Home Office, a judicial review case. When it became clear that the question was what powers the government actually had in a certain situation, the Supreme Court decided that constitutional points should be argued by, effectively, an amicus curiae on behalf of the state, with government lawyers defending their own viewpoint (and other interested parties intervening). This made it, in their view, a case of the state versus the government, with 'Miller' being either an acknowledgement that the applicant remained a party or a means of distinguishing this case from all the other "R. -v- Government" cases over the years, depending on your point of view.
The accepted answer is, unfortunately, quite wrong. In particular, the notation "R (Miller)" does not arise from anything the Supreme Court decided. You can see the first instance judgment is also brought by the Queen.
The modern "application for judicial review" grew out of a process whereby certain courts could issue writs (for example "mandamus" - now a "mandatory order"; "certiorari" - now a "quashing order"; or prohibition). The history is complicated, but these all come to be classified (before the modern era) as "prerogative writs" because they are exercises of the Royal prerogative.
So the process was for the Crown to ask the court to make the writ; but obviously the Crown would usually be acting on someone else's request. In older sources you will see the notation "R v name of minister etc ex parte name of applicant".
You will see that in the High Court, Miller had the claimant "The Queen" and then "on the application of Gina Miller". The Queen is the fictional claimant, but because Gina Miller applied for judicial review. This is abbreviated to "R (Miller)" in short citations.
