The original official code name was the Development of Substitute Materials Project.
But a subsection of the early work (mostly facilities construction) was asssigned to the Army Corp of Engineers under Colonel James Marshal. He set up his headquarters on the island of Manhattan at 270 Broadway. This was for eminently practical reasons as they were using a lot of Columbia University resources. It was also close to the initial primary civilian contractor Stone and Webster. Following ACoE standards the office was titled The Manhattan District.
A lot of the early work was done in NYC, the Manhattan District built at least 10 facilities in and around NYC. Over time the name for the facilities group “Manhattan” kind of took over the whole project. Over time the project ballooned in size, and spread across the country. But by the time places like Chicago IL, Hanford WA, Oak Ridge TN, and Los Alamos NM got involved, the Manhattan nickname was well ensconced, even as the Manhattan became increasingly insignificant in comparison to other sites.