Nagasaki or Fukuoka

We shall be in Kumamoto and we have one day to visit somewhere. We hesitate between Nagasaki or Fukuoka or maybe there any other place recommended?

I personally would definitely make it Nagasaki because the city has an extremely interesting history and a different feel from most other Japanese cities. But many people I know would rather make it Fukuoka because they like large cities and the entertainment and shopping choices that come with them:

The following links introduce both cities in some details:
https://www.japan-guide.com/e/e2161.html
https://www.japan-guide.com/e/e2162.html

Depends on you interests. And mode of transports.

Easy to get to-
Kagoshima
Aso Caldera
Yanagawa

More effort to get to-
Hita
Yufuin
Takachiko

There is a mixture of canals, historic towns, nature, shopping, onsen etc in this list.

Given an apples to apples comparison, I'd choose Nagasaki. Both are good cities to visit, but Nagasaki is a unique city in Japan and not because it was bombed. It has Chinese, Dutch, and Christian influences that make it different from other Japanese cities. It also has some wonderful food like chanpon and saraudon. It is easy to get around to the best sights on its street trolley, and a day pass for it can be convenient and money saving.
Its night view from Mt Inasa is also called the "10 million dollar view" for a good reason.
https://goo.gl/hZVjBi

Fukuoka has its fans too, including me. There is no shortage of diehard fans for its tonkotsu ramen and mentaiko (spiced cod roe). It has some good places, particularly the Nanzoin temple just outside the city, and Tochoji, with the country's largest wooden carved Buddha. If you prefer shopping and nightlife, then Fukuoka easily wins.
https://goo.gl/X3LDaE

I wasnt a massive fan of either and found Nagasaki a little hard to get around being so hilly, but if I had to choose it would be Nagasaki. I really liked Yufuin though, I'd go back there for sure. The Owl zoo was fantastic.

Speaking of Zoos, maybe also look at Nagasaki Bio Park (closer to Sasebo). A bit of a pan to get there, but a lot of fun.

I visited both a few years ago and they have different things to offer.
Nagasaki has interesting history and places to visit, the setting of the city is stunning, surrounded by beautiful mountains and the sea. However, the city itself is quite ugly with many ugly buildings (even by Japan standards) whereas Fukuoka is much bigger, nicer looking architecturally with lovely parks and proper big city benefits.

I'd skip both and go to Mount Aso (if you're not already), it's breathtaking.

You can buy
JR nagasaki slow line ticket 3day =7000 yen
Nagasaki bus 1 day pass=500 yen
Nagasaki tram 1 day pass=600 yen
Sasebo city saihi bus day pass=1000 yen
Sasebo tourist passport=2500 yen

I would pick Nagasaki - I prefer the history and have enjoyed my visits there.
Just on Mount Aso, check the alert levels before venturing - if the exclusion zone is in force or gas concentrations too high, it might be a wasted trip.
https://www.jma.go.jp/bosai/volcano/#type=warning&event_id=503&lang=en

I'd read "The thousand autumns of Jacob do Zoet" by David Mitchell and then go to Nagasaki and visit Dejima (which is where the novel is set).

Fukuoka is great, but Nagasaki is unique historically (and that's even before you think about the atomic bomb museum - which I personally think that everyone who has a chance to visit should go to).

@Winter Visitor

I read that book some years ago and was looking forward to Dejima because of it, sadly because there is no island left with all the land reclaimed from the sea, it was a little disappointing for me.

Worth visiting though.