There's a question asking which answer is correct in "He uses [a car of his son /his son's car]." While I know the right one is 'his son's car', I also remember that I surely heard of 'a friend of mine' somewhere and it was alright to use the 'of+mine' there. So... what makes the former is wrong to use but the latter is alright?
"He uses a car of his son." means the son has more than one car.
"He uses the car of his son". is fine but generally, items that are owned can take the possessive apostrophe s. As in: He uses his son's car. [the son only has one car]
- The son owns a garage and the police found a car of his son [or one of his son's cars] near the crime scene.
Whereas: a friend of mine means: not a friend of someone else (for example, a friend of yours) and is one person in the group of people that are "my friends".
A friend of mine=one of my friends. A car of mine=one of my cars.
My friend [not of mine] is just my friend, not my enemy. There is no idea of group, here.
- A car of mine was used in the drag race and I had not given the driver permission to use it!
A car of mine implies I have more than one car.
A car of his son is a perfectly good, grammatical, comprehensible, Englsh phrase. It's just not one that any English speaker would use in normal circumstances.
However, a car of his son's is idiomatic, and implies that his son may have more than one car. Note that this is parallel with a friend of mine, in that mine is possessive, like son's.