Every time a new pontiff is chosen in a conclave, a senior cardinal goes up to him and asks: "And by what name do you want to be called?"
The question is popped immediately, while all electors are still locked in the Sistine chapel. So the winner had better have done his homework and already picked a name.
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Shortly after, the senior cardinal reads out the pontifical name in Latin from the main balcony of St. Peter's Basilica as part of the "Habemus Papam" -- "We have a pope" -- formula that proclaims the election of a new pope.