"If he chooses the name Pius XIII, it is a clear signal that he
didn’t like Vatican II and wants to move the church backwards," said
the Rev. Thomas Reese, editor of the Jesuit weekly magazine America,
referring to the conservative stance of Pope Pius XII, who died in 1958.
the name John XXIV would signify "a desire to continue the Second
Vatican Council," Reese said. Pius XII’s successor, John XXIII, called
the international gathering of prelates from 1962-65, which was
credited with modernizing the church through its liberalizing reforms.
Of course there are different meanings associated with different names. I also find it interesting that no Pope has ever (and, most likely, will never) chosen the name of ‘Peter’ after Peter the Apostle, the first Pope.
The one name that no pontiff has presumed to duplicate is that of Peter the Apostle, the first pope.
the pope is also known as the "successor of Peter," no one wants to put
himself on the same level as the man who, according to church teaching,
Christ himself put at the head of his flock.
The new Pope is Pope Benedict XVI. Take a look here for more information.