(The recipe)
Let \(\sigma \subseteq \RR^2\) be a two-dimensional cone. Take the convex hull of \((\sigma \intersect N) \sm \ts{0}\). Its compact boundary is a broken line through finitely many lattice points; the rays through those points are exactly the rays of the minimal resolution. Every cone of the refined fan is then smooth, and the resolution is minimal because no exceptional curve has self-intersection \(-1\).
(Degree three)
Take \(\sigma = \Cone\big( (0,1),\ (3,-1) \big)\), whose affine toric variety is the cone over the rational normal cubic and has \(\Cl = \ZZ/3\). The determinant is \(\abs{0 \cdot (-1) - 1 \cdot 3} = 3\), so \(\sigma\) is singular. The only lattice point on the compact boundary of the hull is \((1,0)\), so insert that ray. Check both new cones: \[ \det\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = -1 , \qquad \det\begin{bmatrix} 1 & 0 \\ 3 & -1 \end{bmatrix} = -1 , \] both smooth, so a single blowup resolves. The exceptional curve \(E = D_{(1,0)}\) has self-intersection read from \[ (0,1) + (3,-1) = (3,0) = 3 \cdot (1,0) \rightsquigarrow E^2 = -3 . \]
The same computation for \(\Cone\big( (0,1), (d,-1) \big)\) gives one exceptional curve with \(E^2 = -d\), which is the standard model of the cone over the rational normal curve of degree \(d\): contract the negative section of \(\FF_d\).
For a general cone the boundary lattice points give a chain of rational curves with self-intersections \(-b_1, \ldots, -b_r\), all \(b_i \geq 2\), and the \(b_i\) are the Hirzebruch–Jung continued fraction of the singularity. The \(b_i \geq 2\) is exactly the statement that the resolution is minimal.