Why Fukuoka? Probably because it's the most industrialized, urbanized prefecture on the island and is therefore presumed to have a lot of monied young people eager to escape the big city for a little peace and quiet in the countryside. I say "probably" because I learned about the program from an acquaintance who participated. She went to Gokase with a girlfriend and described it as the best time she's had in a long time.
Participants in the program—which is completely free—spend the daylight hours shopping, strolling, or taking in the town's hot springs. At night, they return to their host family for dinner and perhaps even a trip to a karaoke bar if they're lucky (or unlucky, depending on your opinion of karaoke).
The program is open to any resident of Fukuoka, provided he or she is under 30 years of age. That's because the goal is to attract young people in the hope that they'll not only spend their money liberally (as young folks are known to do) and maybe even decide to live there—like most small rural towns in Japan, Gokase's population is gradually declining, as locals strike out to seek their fortune in the big city.
Another reason the town's population is shrinking is that wives are hard to come by for eldest sons who take over the family farm or shop (and responsibility for caring for aging Mom and Dad). This may explain why Gokase officials place special emphasis on advertising their stay-for-free program to young women. My acquaintance said that one night at her host family's home, a neighbor stopped by to introduce his eligible-bachelor son. Unfortunately for him, no nuptials ensued.
By the way, my acquaintance is 32 years old, but as far as the people of Gokase know, she's 28.