Contemporary readers would probably understand it as an abbreviation for something like Eckford Base Ball Club, of Brooklyn in contrast to "Eckford" clubs in other cities. "Eckford" was not common as the root of a ballclub name — in contrast to "Athletic", "Atlantic", and "Mutual" — so there must have been little need to distinguish the Brooklyn rendition.
The term "Eckford" is likely an abbreviation for a specific local entity, such as the Eckford Base Ball Club of Brooklyn, which serves to differentiate it from similar clubs in other cities.