Do you mean W = span{0,v}? In this case, we do have dim W = 1.

The reason is that the vectors 0 and v arenotlinearly independent (for example, 7*0+ 0*v=0).

Hence span{0,v} = span{v}, and so dim W = 1 (remember that the dimension of a vector space is the number of basis vectors in a given basis, or equivalently, the maximal number of linearly independent vectors in the vector space).

As for why the dimension of the vector space consisting of nothing but the zero vector is 0, you can think of it as a convenient convention. In R^3, the dimension of a plane is 2. Moving down in dimension, the line has dimension 1, and moving down in dimension again, you have the vector space {0}, which is assumed to have dimension 0.