The Collaborative International Dictionary of English v.0.48:

Prerogative \Pre*rog"a*tive\, n. [F. pr['e]rogative, from L.
   praerogativa precedence in voting, preference, privilege, fr.
   praerogativus that is asked before others for his opinion,
   that votes before or first, fr. praerogare to ask before
   another; prae before + rogare to ask. See Rogation.]
   [1913 Webster]
   1. An exclusive or peculiar privilege; prior and indefeasible
      right; fundamental and essential possession; -- used
      generally of an official and hereditary right which may be
      asserted without question, and for the exercise of which
      there is no responsibility or accountability as to the
      fact and the manner of its exercise.
      [1913 Webster]

            The two faculties that are the prerogative of man --
            the powers of abstraction and imagination. --I.
                                                  Taylor.
      [1913 Webster]

            An unconstitutional exercise of his prerogative.
                                                  --Macaulay.
      [1913 Webster]

   2. Precedence; pre["e]minence; first rank. [Obs.]
      [1913 Webster]

            Then give me leave to have prerogative. --Shak.
      [1913 Webster]

   Note: The term came into general use in the conflicts between
         the Crown and Parliaments of Great Britain, especially
         in the time of the Stuarts.
         [1913 Webster]

   Prerogative Court (Eng. Law), a court which formerly had
      authority in the matter of wills and administrations,
      where the deceased left bona notabilia, or effects of the
      value of five pounds, in two or more different dioceses.
      --Blackstone.

   Prerogative office, the office in which wills proved in the
      Prerogative Court were registered.
      [1913 Webster]

   Syn: Privilege; right. See Privilege.
        [1913 Webster]

Bouvier's Law Dictionary, Revised 6th Ed (1856):

PREROGATIVE COURT, eccl. law. The name of a court in England in which all
testaments are proved and administrations granted, when the deceased has
left bona notabilia in the province in some other diocese than that in which
he died. 4 Inst. 335.
     2. The testamentary courts of the two archbishops, in their respective
provinces, are styled prerogative courts, from the prerogative of each
archbishop to grant probates and administrations, where there are bona,
notabilia; but still these are only inferior and subordinate jurisdictions;
and the style of these courts has no connexion with the royal prerogative.
Derivatively, these courts are the king's ecclesiastical courts; but
immediately, they are only the courts of the ecclesiastical ordinary. The
ordinary, and not the crown, appoints the judges of these courts; they are
subject to the control of the king's courts of chancery and common law, in
case they exceed their jurisdiction; and they are subject in some instances
to the command of these courts, if they decline to exercise their
jurisdiction, when by law they ought to exercise it. Per Sir John Nicholl,
In the Goods of George III.; 1 Addams, R. 265; S. C. 2 Eng. Eccl. R. 112.