The earliest popes usually kept their baptismal names. However, later on the Pope typically chooses a new name upon his election as pope. When popes began to take a new name it was to symbolize their shift in ecclesiastical position and to indirectly inform Christendom who in Christian tradition they will try to emulate (or at least a figure they highly respected). "John" presumably indicates St. John the Evangelist, who wrote the Gospel of John. In another example, Pope John Paul I, chose both St. John and St. Paul. Our current Pope chose the name Francis for St. Francis of Assisi, which makes a lot of sense because he was a Jesuit -- a holy order who holds St. Francis in very high esteem.

Now, when you get a series of popes who all choose the same name, part of it is still the original purpose of choosing an important figure to emulate. But now there is also a list of important papal figures who might deserve emulation (e.g. Gregory VII, Leo I, Honorius III). So, not knowing a whole lot about either of these popes, other than John XXII was an Avignon Pope, John XXIII decided the controversy over Antipope John XXIII, and Popes named John have trouble numbering themselves, we can say both probably admired and tried to emulate St. John, which is the likely reason why both chose the name.

Similarly, there was a long time between King William II (d.1100) and King William IV (d.1837) (not counting William III, of Orange); Charles III and Charles IV in France. Sometimes it just happens.

Someone else may know some specific details.