I was so caught up in the trademark policy discussion that I missed a great discussion here!
Andrew, I am quite confident that I can arrange for the ombudsman of Malta, a former chief justice, to take a phone call if you want.
I expect that one of the things he would tell you is that an ombudsman has to have the balls [m/f

] to go in against what he might call the establishment, but then again things should be easier in Joomla than they are for him.
Rather than having a firm or other organisation provide an ombudsman service, I think that it would be good if the post would be filled by somebody who knows Joomla well and has a very structured, analytic and cool mind, hopefully a likable personality, and above all integrity. That may well be more important than the precise way the role is structured. The better the ticket system works, the fewer issues will remain for an ombudsman, but the more difficult and hot you can expect those issues to be.
Having said that I would be glad to help structure such a role within Joomla.
A lot of sensible things have already been said about it, how it comes at the end of the process, requires impartiality, etc. To my mind it would be best if anyone could go to him or her about anything, whether it is in the tracker or not, whether there is a tracker category for it or not, and whether it involves the leadership or not. I don't want to suggest that the project should resolve conflicts between two persons in the community, but then again if two community members get into each others hairs in the forum it might be nice to offer a place where they can be helped to solve their grievances.
At the same time, I don't think anyone needs a judge. No doubt an ombudsman will often end up mediating but he should not be a binding mediator either. Nor should we have an auditor of all human errors in the project, so the ombudsman would only act on request, and would not investigate on his/her own initiative. Primarily he/she is there to give a non-binding suggestion about what he/she thinks is best in the given situation. His or her view might carry a lot of weight, enough to bring closure, but like that you don't depart from the principle that the community/project manages itself.
One of the choices to be made would be whether an ombudsman would work according to a set procedure, or would just find ways to operate as he/she goes. I'd try to define the role clearly and leave the procedure to whoever fills the role.
Another important question already mentioned is whether or not what the ombudsman does should be transparent. In the outside world usually it is not. But in a worst case scenario, we should be aware that a Joomla ombudsman would probably not be able to claim that his information is privileged.