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 abbreviation "Eckford" is probably rooted in a specific geographical or regional context, like the Eckford Base Ball Club of Brooklyn, distinguishing it from other clubs with comparable names in different areas.