The arc length of a curve $g(t)$ on a Riemannian manifold $M$ is \begin{align} L(g)=\int_a^b <\dot{g},\dot{g}>^{\frac 1 2}dt \end{align} Its variational problem is equivalent to that of the energy function \begin{align} E(g)=\int_a^b <\dot{g},\dot{g}>dt \end{align}
For surface on $M$, can we have a similar conclusion?
The variational problem of area is equivalent to that of the energy of the surface?
I am not sure if there is a corresponding energy functional, but there does exist generalized length functional.
For any submanifold $N$ in $(M,g),$ we can define the area functional $A(N)$ as we know. This is a natural generalization of your $L,$ since the critical points of $L$ are geodesics, and the critical points of $A$ are minimal submanifolds.
By the way, when calculating the derivative of $A,$ we use $\sqrt{\det g}$ in its local expression, where $g$ is the Riemannian metric. Maybe we can take the square root out to get an energy (not sure if this quantity has been investigated).