The paperweight is something that he finds beautiful, but doesn't serve an essential purpose. In his society, anything pleasurable is forbiddden and considered dangerous. The paperweight represents his rebellion against this oppression and takes him back to his childhood before the Party took over, when you could have or do something just because it brings pleasure. It also is a symbol of his relationship with Julia, which exists simply to share pleasure and he finds her beautiful. She doesn't serve a 'useful' purpose like providing offspring or a lifetime mate.
To me, the paperweight is sort of symbolic of Winston's hopes and dreams and of his time together with Julia.
Winston buys the paperweight early in the time that he is having the affair with Julia. It is very old and very beautiful. In a way, this symbolizes his hopes -- he wants things to be the way they were before the Party. He wants there to be beauty again in the world.
At the end of Winston and Julia's time together, when the Thought Police storm into the room, the paperweight is broken, symbolizing the end of their time and the end of Winston's dreams.
Winston buys a paperweight in an antique store in the prole district that comes to symbolize his attempt to reconnect with the past. The storeowner describes it as “a beautiful little thing”(84). Symbolically, when the Thought Police arrest Winston at last, the paperweight shatters on the floor. it represented his attempts to finding out the truth to discovering himself but and breaking free of the government, but that broke when the weight shattered.
The coral paperweight is a literalized image of A Memory. The assault upon the human Memory, and its recorded History, is the central attack point of the Party. "Who controls the present controls the past..."
To Winston, this fragile object, a piece of delicate coral from another time, is a poignant metaphor for his own cherished memories, protected within his obdurate mind, as the coral is shielded by the heavy glass, from the intrusions of the thoughtpolice.
When the stormtroopers smash the paperweight, as Winston and Julia are captured, this act symbolizes their future treatment of their flesh, their memories, and the Human Memory itself.
The paperweight is a hint of a forgotten value, lost beauty and the human urge to cherish the perishable.