Follow up question, are there any known living descendants of the Southern court's last emperor?
The answer to this lies in the decision of the Japanese authorities to elevate the status of the Emperor among the public consciousness following the Meiji Restoration in 1868 and use him as a symbol of nationalism and unity for the new nation. To understand how this is related to a 14th century imperial dispute, it's worth briefly going over the history (although it can be fairly confusing for those unfamiliar with it).
When the retired (but head of the Imperial House and therefore de facto reigning) emperor Go-Saga died in 1272, he left it to the Kamakura shogunate to name his successor. The shogunate decided to have Kameyama, the then current emperor, become the new head of the Imperial house. Kameyama abdicated in 1274, leaving the throne to his son, Go-Uda. Kameyama's older brother, the retired emperor Go-Fukakusa, was unhappy with this situation and petitioned the shogunate to have his son named as Go-Uda's heir (this would in turn make Go-Fukakusa head of the Imperial House). The Kamakura shogunate established an uneasy compromise between the two brothers under which the position of emperor would pass between the descendants of Go-Fukakusa (known as the Jimyōin line; the later Northern court) and Kameyama (known as Daikakuji line; the later Southern court) every decade or so.
This unusual arrangement persisted until 1331, when Emperor Go-Daigo (of the Daikakuji line) launched an attempt to overthrow the Kamakura shogunate and restore direct Imperial rule. When the shogunate learned of Go-Daigo's plans, they forced him to abdicate and named Kōgon (of the Jimyōin line) as the new emperor. Go-Daigo's efforts were ultimately successful, however, and in 1333 he deposed Kōgon and restored the Imperial House to political power in what is known as the Kenmu Restoration. Things didn't go very well, however, and by 1336 Go-Daigo had himself been overthrown by Ashikaga Takauji (founder of the Ashikaga shogunate). Go-Daigo fled to Yoshino (what is now Nara Prefecture) and Takauji named Kōmyō (Kōgon's younger brother) as the new emperor in Kyoto. Thus began the Northern (Kyoto) and Southern (Yoshino) Courts period, which would last until 1392 when the Southern emperor Go-Kameyama abdicated in favor of the Northern emperor Go-Komatsu (in exchange for a return to the system of alternating rule between the two lines, although this never happened).
To bring this all back to modern Japanese history, following the Meiji Restoration, the new government (having just overthrown a shogunate to establish what was de jure direct Imperial rule) and its supporters heaped praise upon Emperor Go-Daigo and the Kenmu Restoration and scorned the Northern emperors as willing puppets of the shogunate. For example, Kusunoki Masashige, one of Go-Daigo's doomed generals, was elevated into a paragon of samurai loyalty with tales of his life added to moral education textbooks and a prominent statue placed in front of the Imperial Palace (a statue that also appeared on Japanese banknotes).
But while the state had taken a pro-Southern position for decades, the spark that led to the 1911 decree legitimizing the Southern court came from outside of the government. In 1911, the Yomiuri Shimbun (one of Japan's largest newspapers) published an editorial decrying the fact that the history textbook adopted by the Ministry of Education in 1910 did not take a position on which of the two courts was legitimate. This was seen as undermining the absolute nature of Imperial authority by suggesting that there could ever have been a genuine dispute. This "failure" of the state to properly educate the youth of Japan was further connected by the editorial to the High Treason Incident (a socialist-anarchist plot to assassinate the emperor) of the previous year. The controversy spread to the rest of the Japanese press and was soon taken up as a cudgel that the opposition parties could use to attack the government in the Diet. This in turn led the Katsura government to issue a cabinet decision officially recognizing the Southern Court as legitimate.
Source: John S. Brownlee, Japanese Historians and the National Myths, 1600-1945
Follow up, who was it that made this determination? How did they come to their conclusion?