Monaco's Minister of State is usually a French civil servant, who often lacks prior ties to the principality. Is the Minister of State awarded citizenship in Monaco by virtue of office? Or does he or she remain solely a French citizen?
He or she retains their citizenship.
Monaco only grants citizenship to those whose father is a citizen of Monaco, (or mother in the case of illegitimate children whose father is unknown). It does not allow dual nationality. The Minister of State is appointed by the Prince, from a short-list of candidates selected by the French government. The Minister of State serves for five years (previously three years) then steps down, and typically returns to France, though previous Ministers of State have maintained interests in Monaco, for example by becoming directors of Monegasque companies.
Wikipedia notes that, prior to the 2002 amendments to Monaco's constitution, the Minister of State was required to be a French National. The 2002 amendments removed that restriction, and Monacan Gilles Tonelli served as acting Minister of State, during Michel Roger's illness.
The relationship between France and Monaco developed as a consequence, first of Medieval notions of suzerainty, then as a pawn in 18th and 19th century European politics, and finally in terms of treaties, notably that in 1918, which established Monaco as a client state of France. The right of France to nominate the Minister of State is part of this unequal relationship between France and Monaco.
That a French citizen may be a senior member of the executive of Monaco may seem unusual, but it is a fairly common form of government in British Colonial territories. For example, the Governor of Bermuda is typically a British diplomat, even the Governor General of Canada had always been a member of the British aristocracy, prior to 1952.