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.