The explanation that “name for ourselves” is equivalent to “reputation” is unsatisfactory. “Reputation” is important in relation to others outside the city. “Name for ourselves” has probably the intention of impressing the city’s own inhabitants, to show the pride, the power, and the impressiveness of the city and the tower.
4 Then they said, “Come, let us build ourselves a city and a tower with its top in the heavens, and let us make a name for ourselves, lest we be dispersed over the face of the whole earth.”