I run a WSS3.0 and Moss 2007 site collection. I recently created one
particular team site for a department and assigned an owner to it.
Everything works fine except one tiny tiny thing. The owner can do
everything on the site except enter new members to the site.

When he creates a group, then tries to assign a member to it, he does
not see the NEW USER selection - only the NEW GROUP selection. This
is no mater what group is created.

What am I missing here?

All my other sites with owners do not have this trouble.

This site is nothing more than a Team site, with unique permissions
that do not inherit permissions from the parent site. I create sites
like this all the time, but for some reason this one won't allow its
owners to add a member.