I'm not sure how the system works exactly, but take a look at this list of New Hampshire towns:

https://en.wikipedia.org/wiki/List_of_cities_and_towns_in_New_Hampshire

The word 'Center' occurs in a town name 40 times. On the map, we see Strafford, Center Strafford and Center Barnstead outside of Rochester. I think it's likely that local people simply called the nearest town with a name like that 'Center.'

Further north, we find Grafton, East Grafton and Grafton Center. For some reason, these towns must have been linked in people's minds.   

This isn't just a New England phenomenon. Wisconsin has Richland Center and Alma Center.