I like the idea of having a lawyer in the department oversee the relationship with a law firm that handles significant amounts of work (See my post of May 18, 2007: inside counterpart appointed for each major firm; Aug. 18, 2008: BT and its inside relationship partners for key firms; and July 8, 2009: pros and cons of a responsible lawyer in-house.). At first blush, you might want the general counsel to be the point person, but I think that is a mistake in departments of medium size and up. The general counsel has more important obligations than the smooth running of relationships with key firms.
How should you choose other lawyers for the role? It is a close call whether you want a coordinating counsel to be a lawyer who has worked extensively with the firm. In general, this makes sense because that lawyer knows more about the firm and its lawyers. The drawback is that the lawyer might lack objectivity.
The person needs to be senior enough to have gravitas and an ability to convey criticisms gracefully and effectively. Likewise, other lawyers in the department must trust that person to keep confidences.
The person needs to care about doing the job well. It is, after all, administrative work, but someone needs to serve as coach, referee, and parent.