In a legal system that prizes the principle of stare decisis—the upholding of prior judicial opinions if doing so is possible without offending fundamental notions of justice—the crucial words "cold finality" refer to the authoritative force that such judgments acquire when they are issued and upheld on appeal.
Supposing that the limits of appeal in Oliver v. City of Raleigh (the case whose majority opinion North Carolina state Supreme Court Justice Heriot Clarkson was dissenting from) had been reached, the majority opinion became part of the established law of North Carolina—as impervious to change (owing to the force of judicial precedent) as the cold, marble columns in a courthouse's architectural façade.
The point of Clarkson's dissent is threefold: to explode any notion that the court was unanimous in its reasoning and conclusions; to provide a line of analysis that some future court (he hopes) may find so persuasive that it concludes that the majority opinion was simply wrong; and to influence the outcomes of similar cases not governed by North Carolina precedent. In the United States' form of federalism, state courts may (and often do) consider the rulings and opinions of courts in other states if those decisions are relevant to the case at hand, and if their state doesn't already have a controlling precedent of its own. Versions of Clarkson's three motives underlie virtually all dissents in U.S. state and federal district courts.
Just for fun, I ran a Google Books search for "cold finality of" to see what things are most often introduced with that phrase in published works of the past. The three most common are "the cold finality of print/type/the printed page"; "the cold finality of his/her voice/tone"; and "the cold finality of death/the tomb."