Right, prerogative, privilege, perquisite, appanage, birthright can all mean something to which a person has a just or legal claim.
They differ in their implications both of the nature of the thing claimed and of the grounds of the claim.
Right, the most inclusive term, may designate something (as a power, a condition of existence, or a possession) to which one is entitled by nature, by the principles of morality, by grant, by the laws of the land, or by purchase.
Prerogative denotes a right which belongs to an actual or a legal person by virtue of status (as in sex, rank, office, or character) and which thereby gives precedence, superiority, or an advantage over others.
Privilege applies to a special right either granted as a favor or concession or belonging to one as a prerogative; privilege often implies an advantage over others.
Perquisite signifies something, usually money or a thing of monetary value, to which one is entitled, especially by custom, as an addition to one’s regular revenue, salary, or wages.
Appanage is often used as if it meant merely an adjunct or appurtenance, but more precisely it can denote something to which one has a claim through custom, through tradition, or through natural necessity.
Birthright, which basically applies to the property or possessions belonging to one by right of inheritance (see HERITAGE ), has acquired extended use in which it differs from right only in being restricted to a right to which one is entitled by some reason connected with one’s nativity (as by being a man, a native born citizen, or a descendant of a particular family line).