The problem is that you have a triangle"triangle fan": A bunch of triangles converging on a single vertex (aalso known as a pole). The subsurf modifier will not work well with that. You
You need quads.
Delete the vertex at the center of all the triangles.
Select the inner ring loop and press E to extrude and without moving the mouse press escape
Then scale all of the vertices to 0 (S0)
That will make all of the newly created vertices converge in a single place. And the subsurf should work well.