Generating unique vertices for a procedural mesh

As the predefined plain mesh is not intended for such a usage, I think you will need to generate your own mesh. This will mean not much more effort for you, because you are already manipulating the vertices yourself. As long as one triangle has different normals at its corners, they will be interpolated per fragment. That's how the pipeline works.

Here is the link from Unity Manual that you might already know: GeneratingMeshGeometryProcedurally

It is really simple: Create vertices, create triangles by giving the indices of 3 vertices each and creating the normals.

Even though this thread was resurrected a bit late I'll leave this here for anyone interested. To create a procedural plane useful for terrain generation it's easiest to iterate over the vertices and create quads of two tris each for every vertex. One way looks like this:

 public void CreateMesh(int size)
     {
         List<Vector3> verts = new List<Vector3>(); // Index used in tri list
         List<int> tris = new List<int>(); // Every 3 ints represents a triangle
         List<Vector2> uvs = new List<Vector2>(); // Vertex in 0-1 UV space
         for (int i = 0; i < size; i++)
         {
             for (int j = 0; j < size; j++)
             {
                 verts.Add(new Vector3(i, 0, j)); // Add noise function into y
                 uvs.Add(new Vector2((float)i / size, (float)j / size));
                 if (i == 0 || j == 0) continue; // First bottom and left skipped
                 tris.Add(size * i + j); //Top right
                 tris.Add(size * i + (j - 1)); //Bottom right
                 tris.Add(size * (i - 1) + (j - 1)); //Bottom left - First triangle
                 tris.Add(size * (i - 1) + (j - 1)); //Bottom left 
                 tris.Add(size * (i - 1) + j); //Top left
                 tris.Add(size * i + j); //Top right - Second triangle
             }
         }
         Mesh mesh = new Mesh();
         mesh.vertices = verts.ToArray();
         mesh.uv = uvs.ToArray();
         mesh.triangles = tris.ToArray();
         mesh.RecalculateNormals();
         
         GameObject terrain = new GameObject("Terrain");
         terrain.AddComponent<MeshFilter>();
         terrain.AddComponent<MeshRenderer>();
         terrain.GetComponent<MeshFilter>().mesh = mesh;
         // Load a material named "TerrainMat" from a folder named "Resources"
         Material terrainMat = Resources.Load<Material>("TerrainMat");
         terrain.GetComponent<Renderer>().material = terrainMat;
     }

The thing to remember about generating meshes procedurally is that the order of the vertices matters and the triangles have to be made clockwise. If you're interested in other applications of procedural meshes you can still use this method but switch out the i and j variables in the for loops or in the vertex declaration. Besides the terrain method using noise you can also get things like curves, circles, and even spheres.