In modern English, of is not commonly used to indicate possession, which perhaps explains Jason's comment that it feels biblical. Relationships are (generally) mutual so examples using relatives make this a little more confusing than it needs to be. If we instead think of possessing an item, like a car, a native English speaker would find the sentence "This is the car of John" strange, archaic or maybe even wrong.
My native feeling for the word of in your question, is more similar to "one of many", as in, she has many cars. "This car is one of hers." Could be short for "This car is one of her cars."
As a final example, let's consider the phrase "a part of her" and compare with "a part of hers". The first phrase says she is composed of parts and we are talking about one of them. The second says that she possesses parts but it's unclear if they are car parts or some other kind of part (like a part of her personality).