To create a partial answer, eigenvalues of banded Toeplitz matrices accumulate on some real algebraic curves in $\mathbb{C}$, (this has been proved by Schmidt and Spitzer around 1970.). Now, if we also attach a point-mass at each eigenvalue, (for fixed matrix size) with equal mass at each point, and with total mass one, we get a probability measure. These measures converge in a certain sense to some limit measure, see for example the book by Bender and Böttcher .
Now, the sum $c_n = \frac{1}{n} \sum_{k=1}^n |\lambda_{n,k}|$ can then be interpreted
as the center of mass mean distance to 0 of all the eigenvalues from matrix of size $n \times n.$
Say that the associated point measure is called $\mu_n$ (mass $1/n$ at each eigenvalue),
then we know that $\mu_n \to \mu$ for some $\mu$ in some sense.
Note that $c_n = \int_{\mathbb{C}} |z| d\mu_n(z)$ and then what you are looking for is $c = \int_{\mathbb{C}} |z| d\mu(z)$.
As an explicit example, taking your matrix to be tridiagonal, will give rise to characteristic polynomials which are also a family of orthogonal polynomials, w.r.t some measure (the limit of point measures, actually), see e.g. this paper i just googled.