Minamata is above all known for one thing: the contamination of its bay with mercury from a chemical plant and the resulting Minamata disease (水俣病 Minamata-byō) that resulted neurological symptoms including paralysis, insanity and death. The culprit, Chisso Corporation — the town's largest employer by far — did all it could to avoid blame, including suppressing internal research showing that its wastewater caused the disease in cats and later claiming it had cleaned up its wastewater in 1959 while in fact it hadn't, leading to an epidemic of severe birth defects. When the second wave of cases broke out, photographer W. Eugene Smith brought the disease to the world's attention in 1972, and was severely beaten for his trouble by yakuza goons. In the end, an estimated 17,000 people were affected.
Given that sad history, the town is nowhere near as grim as you might expect: it's on Kumamoto's scenic Shiranui Coast, occasionally dubbed "the Mediterranean of Japan", and there are quite a few hot springs nearby.
Minamata is served by Shin-Minamata station on the Kyushu Shinkansen line from Fukuoka (2 hrs), Kumamoto (40 min) and Kagoshima (35 min). The cheaper but slower option is to use the private Hisatsu Orange Railway from Yatsushiro (1 hr) to Satsumasendai (90 min), which stops at both Shin-Minamata and "old" Minamata station.
The hot springs towns of Yunoko Onsen (湯の児温泉) and Yunotsuru Onsen (湯の鶴温泉) are nearby.