Youngest Since 1400
John Paul II (#265) was 58 years, 3 months & 4 weeks old when elected pope on October 16, 1978. He was the youngest pope to be elected during the twentieth century, Benedict XV (#259), elected in September 1914 having been 531 days older.
John Paul II was, however, the twentieth youngest pope to be elected as of 1400 [dates pertaining to the popes prior to 1400 are either unreliable or unavailable and as such are impractical for meaningful analysis].
Benedict XV was the 22nd oldest [again as of 1400].
In June 1846, Pius IX (#256) was elected at the age of 54. Two popes separated Pius IX from Benedict XV. Pius IX was the 12th youngest since 1400.
Average age at election of the 62 popes elected since 1400 is 62.4 years.
Per my research it would appear that the minimum theoretical age at which one could become pope, in the future, is 25 – that being the minimum age to be a Catholic priest or deacon.
THE YOUNGEST EVER
One of the youngest popes ever was probably John (‘Octavian’) XII (#131), the illegitimate son of Alberic II who ruled Rome from 932 to 954. Alberic, on his deathbed, coerced influential Romans to promise that they would make sure his son, Octavian, would succeed him as the ruler of Rome and also be appointed the next pope. Octavian became John XII [his step-uncle having been John XI (#126)] in December 955 when Agapetus II (#130) died. John was supposed to have been around 18 years of age at that point.
The infamous Benedict IX (#146, #148 & 151), who served an unprecedented three terms as pope, was also quite young when first elected in October 1032. He was the last layman to be elected pope. Though there are those that claim that he was but a teenager when elected in reality he was probably in his twenties.
THE TEN YOUNGEST POPES SINCE 1400
The youngest pope elected since 1400 was Leo X (#218) at the age of 37, in 1513. He the second son of the famous Lorenzo ‘il Magnifico’ Medici of Florence was created a cardinal, albeit without it being publicized (though this was prior to the in pectore practice that came to be in 1536) when he was but thirteen. Leo X, a cardinal deacon when elected, also happens to be the last non-priest/monk to be elected pope. It is said that on being elected he told his retinue ‘God has given us the papacy. Now let us enjoy it.’ Alak, this was not to be the case. His papacy was majorly buffeted by the rise of Martin Luther’s Reformation. He would die of malaria close to his 46th birthday.
The second youngest, since 1400, happens to be Leo X’s cousin Clement VII (#220), one pope later, at the age of 45. [So there is a 8 year difference between the youngest and the second youngest.] Clement VII’s parents were not married making him the last known pope of illegitimate birth.
The ten youngest popes, since 1400, to be elected are:
Note that three of these popes, #207 to #209, were consecutive; i.e., three popes in succession elected prior to their fiftieth birthday.
It is interesting that the six popes who were elected before they turned 50 did not enjoy particularly long pontificates. Eugene IV, who died at 62, was to be the oldest from this group. During the last 150 years the cardinals have hesitated about electing young, i.e., those in their fifties, given the possibility of a pontificate that might last three decades.