Set elements are constant and may not be modified in place. Modification could change the ordering predicate without changing the position of the element which would violate the constraints of the data structure.
However now in the future (C++17), modifying the element without erasure is possible with the extract
member function:
std::set<std::string> stringset{"a", "b", "c"};
auto node = stringset.extract(stringset.begin());
std::string& str = node.value();
str = "d";
stringset.insert(std::move(node));
There is still the cost of list operations but the object itself is not destroyed or copied.