Jamie,
This is quite interesting. This could be formulated as a classic location-allocation problem type. That is, given a set of k candidate locations to choose from, decide which j (< k) locations (in your case, j=1) are best suited to "service" a set of demand points, such that each demand point gets allocated to its nearest chosen location.
There are several ways to formulate the objective function for solving such a problem (e.g., minimize the total weighted impedance to the demand locations, maximize the number of demand locations which are within a specified coverage impedance, etc.). So far we have been discussing this problem in the context of finding the single junction (the "service" location) which minimizes the maximum impedance to all other junctions (the "demand" locations). However, we could easily reformulate the problem to choose the junction which minimizes the total (and thus, average) impedance to all other junctions. This helps to eliminate the effects of having one or two outlier demand locations. The point is that there are many available options for defining your objective function for this problem type.
We have developed a new solver in Network Analyst at the 10 release to solve just this problem type: the Location-Allocation solver. Its capabilities are much more advanced than I have even attempted to describe here, with many more options for greater flexibility in defining your objective function for selecting/locating your facilities.
I would imagine if your tri-county region is serious about establishing a proper location for their distribution center, then they have to consider the possibility that they might not be able to simply establish such a facility at any possible junction in the network (e.g., perhaps there is no available real estate in a given area of your network coverage). Therefore, using the Location-Allocation solver, they could effectively specify which candidate locations (even ones located midway along edges) that they are willing (and able) to select from, then specify the number of the distribution centers they are planning to establish (e.g., 1, so far in our discussion), and then specify exactly (and only) those locations which the distribution center would truly be servicing (this may or may not include all junctions of the network; there may be a small number of explicit locations to be serviced by this distribution center).
Please stop by the island at the UC and we can discuss the options in further detail.