Well, if we break it up into parts, i.e. 二 = ni, 日 = day ...but 二 and 日together = futsuka, which means "second-day". 市 usually means city however in this instance it means "market."
Therefore, as Sherlock Holmes would say,
However, there is a tangible element to it. Hakata is the Sea Port of Fukuoka, and as such, much trade would be conducted there. In time, market towns began to spring up, to capitalise on their location between Sea Ports and the more remote villages. They would often take the name from which day of the month they would open. For example, if as in Toyama Prefecture, Mikkaichi was so named because it was the third day of the month they opened. Mie Prefecture has Yokkaichi, and so on. Futsukaichi is just the same, a market town that opens on the second-day of the month.
Nowadays, little of this tradition remains, except... Aeon Mall is definitely the market of choice for the Futsukaichi public and people from all the smaller villages come to Aeon's massive market. It is open everyday but if you can imagine what it would be like if it was open just for a short period, on the second day of the month - I wonder if that is what it would have been like long ago in Futsukaichi?