Monte Carlo is not an old name. It is a prince's name, and it is only about 160 years old.
The Principality of Monaco's official history site says Prince Charles III gave his name to the new town growing on the plateau of the Spélugues in 1866. Britannica agrees that the settlement around the casino was named Monte-Carlo "in honour of the prince."
The town grew around the casino and the Hôtel de Paris, which the history page says had opened two years earlier. The government's place page says Charles III thought gambling could attract, as in Germany, a wealthy clientele. It adds that in 1866 the headland "was baptised with a name that promised a special future: Monte-Carlo."
Britannica says the prince built his first casino in 1865. By 1869 he was able to abolish taxation in the principality. The present casino building dates from 1878 and was designed by Charles Garnier, the architect of the Paris Opéra.
The name itself is plain. Monte is the Italian word for mount, and Carlo is Charles, so Monte Carlo means "Mount Charles." Etymonline renders it "Charles's Mountain" and dates the town's founding to 1866.
The name later travelled far beyond Monaco. Los Alamos National Laboratory says Nick Metropolis coined the name of the Monte Carlo method shortly after a 1947 letter from John von Neumann, because of the method's probabilistic nature. The laboratory adds that the name also nods to Ulam's Polish uncle, who borrowed money from relatives because he "just had to go to Monte Carlo."
Monte Carlo is part of Monaco rather than a separate country. The government place page describes it as a headland where the principality built its bathing and gambling establishments.