Is it okay to suggest a Professor Emeritus as a mentor when applying for US mathematics postdocs?
Here is roughly the way it works in my department, and I imagine this is fairly typical for math research departments in the US.


People apply for postdocs.
The faculty review applications.
Faculty who are interested in certain candidates push for them.
Some committee and/or the chair decides on a ranking of the candidates individual professors are excited about.
Offers are made.


For 4, typically getting a postdoc for an emeritus professor will have very low priority, and unless the emeritus prof is active, they probably wouldn't push for you anyway.  I believe that emeriti typically don't even have permission to look at applications.

However, I think it's a little weird that they require you to name one faculty with common interests.  I don't remember ever seeing this when I applied.  (Is it exactly one?  If not, you could mention regular and emeritus faculty.  If so, maybe that is to assign that faculty member to review your application.)  My suggestion (say if you can only name exactly one) is to choose a regular (non-retired) faculty member, but possibly mention the emeritus professor in your cover letter.  If you think the emeritus professor is still active and would be interested in having you there, maybe send them an email to let them know you applied.  Then they might discuss your application with other faculty.