"Mother Church" is not simply an affectionate name for the oldest or largest Christian Science building. It names a specific constitutional relationship laid out in the Manual of The Mother Church, the governing bylaws Mary Baker Eddy wrote and continued to revise until her death in 1910: every other Church of Christ, Scientist anywhere in the world is organized as a branch of this one church in Boston, and none of them, including the Mother Church itself, has an ordained minister.
Services are led instead by two Readers, a First Reader and a Second Reader, who read aloud from the Bible and from Eddy's Science and Health with Key to the Scriptures according to a weekly lesson-sermon set centrally for every branch church to use on the same Sunday, so that a Christian Science service in Boston and one on the far side of the world are, on any given week, reading the same passages. The Mother Church itself has no additional authority over a branch church's local property or membership; what it holds is the constitutional center of a structure without ordained clergy, the source of the shared lesson-sermon, and the seat of the Christian Science Board of Directors the Manual establishes to govern the denomination's central affairs.
That arrangement, uncommon even among churches that reject a professional priesthood, is why the building's common name describes a relationship rather than merely a rank: every branch traces its authority back to bylaws issued from this one address.