Well, of course there are ways to rephrase most anything. I am, however, interested in whether there's a way to express the "a relative of one patient" notion through the possessive 's.
A related question is whether a native speaker would be sure that one patient's relative necessarily means the relative, or he would be ambiguous whether it means the relative or a relative.
In a specialized context (such as among people who work at a hospital), "patient's relative" could conceivably become a set phrase, in which case sentences such as "there are some patient's relatives waiting outside" would become possible (contrast * "there are some Greg Egan's stories on the shelf").
This is presumably what happened with "girls' school". Very rarely, it can even happen with proper nouns, as in the mathematical term Green's function. But this is not part of the syntax of the possessive ; it is the re...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Of course, for "every Monday", the last one should have been dated July 22-28. *cough*