I think you would be pretty happy with the Willmore functional for, well, compact orientable
$C^\infty$ surfaces in $\mathbb R^3.$ It is just the integral of the square of the *mean* curvature or
$$ \frac{1}{2 \pi} \int_{M^2} \; \; H^2 \; dS  $$
This quantity is at least 2, and is only equal to 2 for a round sphere. The Willmore Conjecture is that the minimum for an imbedded torus is achieved on the torus created by revolving a circle of radius 1 with its center at distance $\sqrt 2$ from the axis of revolution. Here the functional has value $2 \pi^2.$ Leon Simon proved that the minimum (a priori the infimum) is achieved. Rob Kusner found some rather earlier references (before Willmore) to this problem. $$ $$  See, for example, "Total Curvature in Riemannian Geometry" by Thomas J. Willmore. $$ $$ I do not expect there would be much trouble making a discrete version of this.