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 $ \pi.$ 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.
$$ $$
NOTE: sometimes Willmore writes with the $2 \pi$ divisor, sometimes not.
$$ $$
I found a nice wiki page and some pdf's with references and other information, one a proposal for a workshop at Oberwohlfach. Anyway,
http://en.wikipedia.org/wiki/Willmore_energy and 
http://www.mfo.de/programme/schedule/2010/43b/programme1043b.pdf   and  
http://www.warwick.ac.uk/~maseq/wmsri.pdf   and 
http://www.math.ethz.ch/~riviere/papers/riviere-tartar.pdf