The Argentine-born pope presented the queen with a stone orb topped with a sharp cross, saying it's for "el ninetto" — a term of endearment for little child. Elizabeth said George "will be thrilled by that — when he's a little older."
The exchange of gifts capped a private meeting of just under 20 minutes between Elizabeth, her husband, Prince Philip, and Francis — with communication facilitated by an interpreter.
Francis did say "Welcome!" in English when the royal couple arrived about 20 minutes late for their appointment.
"Sorry to keep you waiting," Elizabeth told Francis. "We were having a very pleasant lunch with the president." She was referring to Italy's head of state, Giorgio Napolitano, who hosted the queen and Philip for lunch at the presidential Quirinal palace.
The white-gloved monarch arrived earlier in the day in Rome wearing a lilac-colored spring coat and hat that matched the wisteria blooming over much of the Italian capital.
Previously, Elizabeth had met with four pontiffs, starting with Pius XII in 1951, a year before her accession to the throne.Elizabeth is head of the Anglican church. There was no immediate word on what was discussed in the meeting with Francis, who is in his second year as pontiff.
The queen's day in Rome was a makeup visit. Illness had forced her to cancel a 2013 trip to Rome. "She doesn't like leaving obligations unfulfilled so she was determined to reinstate that visit," Britain's ambassador to the Holy See, Nigel Baker, told Vatican Radio.