I was curious myself about the answer, so I searched around on the Internet. A blog called "Nancy's Baby Names" has an entry about how Pio (Pius) was the name of the first baby born (in 1929 to a servant) in the newly formed state.
Also, it's possible a baby could be abandoned there. That came to mind when I read about baby hatches in Wikipedia: "In December 2006 a modern hatch was installed at the Policlinico Casilino in Rome and in February 2007 it received its first abandoned child. There are also plans to install one at the Santo Spirito hospital at the Vatican City, the home of one of the original foundling wheels."