Step 1: Idea:
Ask which choice is an intrinsic property.
Step 2: Result:
Area, mass and length describe the size of a rod, and doubling any of them does not change what the rod is made of. $K$ stays the same for a thick or thin copper wire. Only the material matters.
Final Answer:
Thermal conductivity depends on the material of the rod, option (C).
\[ \boxed{\text{Material of the rod}} \]