I particularly like this question, it is a question philosophers have debated for ages. It's also important that I mention the question is related to Theseus's paradox, often called the ship of Theseus. In your case, the object is not "named." Also, it's more of a philosophical question than a science question.
A popular philosopher, Aristotle, believed that there are several reasons that would determine if an object changed piece by piece would remain the same. He made mention of the reason the object originally has that form, what makes up the object piece by piece, why the object was created and the process of creating the object.
The answer to your question depends on a person's perspective. After changing an object piece by piece it remains the same design, but the materials making that object is not the same materials originally used. For example, a car would still be a car, when all its parts are changed. The concept is still the same, but the materials are now different.
Would you judge an object based on an object's concept or by its materials?
That is what determines if the object remains the same. Some people may believe that every object that has the same concept is the same object. While others believe that any object that the materials are the same, unreplaced, is the same object. Basing your judgement on its materials, means giving attention to its physical attributes. So the options are actually between concept vs physicality. Aristotle picked concept, but I would be more interested in physicality.
Nevertheless, none of the choices would be wrong, it's entirely based on perspective.