Given a $p\times p$ matrix $A_p$, the average eigenvalue can easily be computed by $$ave_i(\lambda_i(A_p)) = \frac 1p\operatorname{tr}(A_p).$$
Suppose that we have a sequence $A_1,\dots,A_p,\dots$ such that $A_p$ converges to a limiting spectral distribution compactly supported in $(0,\infty)$. The definition of the spectral distribution of a matrix $A_p$ is $$s.d.(A_p) = \frac 1p\sum_{i=1}^p \delta_{\lambda_i(A_p)},$$ where $\delta$ is the dirac-delta the assumption above means that as $p$ goes to infinity this converges to a limiting distribution with compact support.
I believe in this case the notion of "average eigenvalue" is just the expected value of the distribution.
However, for the task I am doing it would be useful to be able to think about this limiting distribution as an operator itself. Does this notion make sense? If so, what properties would this operator have that I can exploit? How would the notion of "average eigenvalue" make sense for this operator?
Edit: (the text below was an answer to the original version of the question)
You say "intuitively it is obvious that a notion of average eigenvalue should exist". I think, on the other hand, that it is obvious that there is no notion of average eigenvalue.
First of all, your operator may lack eigenvalues at all. So, if $A$ has no eigenvalues and its spectrum is $[1,2]$, what's the "average eigenvalue"?
You could say that you could simply average the spectrum geometrically. Even then, $A$ may have part of the spectrum made up of eigenvalues and part not eigenvalues. You could decide that infinite multiplicitly always trumps finite multiplicity, so you only consider the essential spectrum. Even them, if the spectrum is $$\left\{1,\frac43\right\}\cup\left[\frac53,2\right],$$ what would the "average" be? There will always be one more pathological case to deal with.