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 <a href="http://www.amazon.com/Spectral-Properties-Banded-Toeplitz-Matrices/dp/0898715997">Bender and Böttcher </a>.

Now, the sum $c_n = \frac{1}{n} \sum_{k=1}^n |\lambda_{n,k}|$ can then be interpreted 
as the center of mass 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. <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDkQFjAA&url=http%3A%2F%2Feuler.us.es%2F~renato%2Fpapers%2Fchain-revised-1.pdf&ei=iTrpUIi5MaOH4ASojYDADQ&usg=AFQjCNGMKq-EYEq5rqZeXhVcv4eqrHj2Lw&bvm=bv.1355534169,d.bGE&cad=rja">this paper i just googled</a>.