Slag:
Sorry for the hairdo comment!
More seriously, my experience with this topic is limited, but I would guess a large percentage of project sponsors fit into two categories. Either they really have one particular architect in mind or they create a "short list", maybe three to four candidates.
I can't imagine interviewing more than three or four, especially if serious thought has been given to creating the short list in the first place.
Now, what about criteria? I would have to think a great deal depends on the type of project. One of the "brand names" probably has a big advantage where there are serious marketing issues at stakes, e.g., real estate or high priced membership. The choices open up where other project objectives have greater priority, e.g., building a certain quality course or working within a given budget.
Were the decision mine, great weight would be given to the overall project vision and what kind of course best fit within that vision. I think you want to hire someone with whom you share common values and architectural interests. Keep in mind that for many project sponsors, the project may be the only course they build, a once in a lifetime experience. You really do, then, want to feel comfortable with the individual you hire.
FYI, Dusty Murdock once told me that he considered only two architects for the Sand Ridge project: C&C and Fazio. In the end, marketing concerns and project management expertise gave the nod to Fazio.