Honestly I don't find many interesting places at Fukuoka. Just use Fukuoka as a base for you to travel out to other nearby cities, counties. Hakata station is a major station for many shinkansen lines and you can travel to Kumamoto, Yufuin, Beppu, Oita, Nagasaki for instance.

I just came back from Northern Kyushu recently and this is my personal preference, of course everyone has different opinions.

Depends on what ur party would like to do, see, visit, experience.

You could do Dazaifu+Kyushu museum in half to 3/4 a day, Nanzoin in half a day, Ohori park in half a day, Sumiyoshi, Kushida shrines and other nearby temples & shrines in 3/4 of a day or the whole day if you include the nearby museums, then remainder can be shopping in Tenjin area, Amu Plaza, or Kamikawabata shotengai or all 3. Since you have elderly, take more time at each places.

Or else, can go other prefectures as fruitie mentioned. Nagasaki is just a 2 1/2 hr bus ride away, if ur party is into the A Bomb thingy among many others. The 3 best views of Japan is in Nagasaki and best viewed at nite but during winter, temps might be unbearable for some. Nagasaki is also less popular with PRCs so it feels less touristy and more authentic feel, until a cruise ships docks on weekends. lol!