A solid that has only one vertex is
Cone
Pyramid:
A Pyramid has only one base and all triangular lateral faces as shown in the figure.
The number of vertices, faces, and edges depends on the polygonal base.
All the pyramids will have definitely more than one vertices.
For Example:
Square pyramid: A pyramid with a square base is called a square pyramid. It has faces, edges and vertices.
Cube:
A Cube is a three dimensional object with faces, edges and vertices.
Therefore, a cube also has vertices more than one.
Cone:
A cone is a solid figure with a circular base, one curved surface.
It has only one circular edge and one vertex.
Cylinder:
A cylinder is a solid figure with two identical circular bases and one curved surface.
It has two circular edges and no vertices.
Among all the figures, Only a cone has one vertex.
Therefore, The correct option is (C).