There are no criteria that are “set in stone,” so to speak, but a pope usually chooses to honor a predecessor or someone he otherwise admires (e.g., John XXIII sought to honor his father); or he chooses to give an indication of the course of his papacy (e.g., Pius XII sought to indicate that he would continue Pius XI’s work).
DebbieB:
Are there restrictions on the names he can choose?
No. However, it is an unwritten custom that popes do not choose the name Peter. There is no rule against such a choice, but it is generally considered a matter of prudential judgment for popes not to invite comparisons between themselves and the first and original pope Christ himself chose.
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