"One of my friends" vs "a friend of mine"

There is no difference between the two constructions in most cases as far as meaning goes (OALD); this is confirmed by the following ELL sources.

question, its accepted answer; it is useful to read all the answers to that question.

The construction is also used widely for the following words, as this ngram shows.

acquaintance, words, cousin, fault, son, business, letter

                                      

Plenty of other words can be found, so that there is none that seems to be excepted from this usage: job, book, car, girl, shirt, trousers, socks, pen , pencil, fate, etc.

One difference is found relative to the grammatical environment where the construction is found.

For instance "This, my red car, is the only vehicle I drive." is not a well established form, it is at best found very colloquial (user LPH's appraisal of the register). If the comma is not used (making the pronoun "this" into a determiner) this sentence is in no way correct.

From the following source, Google Books, one sees that the construction "of mine" is helpful when the combination of the indefinite article (also "another") or of a demonstrative and the possessive is wanted (there are other cases: after "any" and "what" for instance).

This source confirms further that the structure "of mine" is not normally used unless there is "a post-head adjunct".

LPH's examples

• The red shirt of mine was torn to pieces. not likely; instead, "My red shirt was torn to pieces."
• The red shirt of mine in the washing machine is torn at the elbow.
• A red shirt of mine (that I wouldn't wear at all, not even for doing the dirtiest work,) was on the floor.
• This red shirt of mine is so old that I think I'll never wear it again.

Google Books

ex. 1 It had been approximately one week since I first began the massive undertaking of building this soapbox derby-car of mine.

ex. 2 It was such a beautiful machine, this car of mine.

ex. 3 I think my new friends could see how frustrated I was with this new job of mine, I really didn't fit it and I started to wonder if I had made a huge mistake by leaving the community.

ex. 4 Another job of mine was cleaning the walk-in oven.

ex; 5 …and then had gone to Editha and my own supper (waited on by many a wondering, kindly one of these corded, russet Brothers), when that strange fate of mine overtook me once again.

ex. 6 By what singular fate of mine , conscript fathers , shall I say it comes to pass , that for these twenty years there has not been an enemy to the public

ex. 7, an exception I mean, I was no longer driving the piece-of-junk Japanese car of mine;

A second difference is found relative to the set of elements out of which one is selecting one or more. This point is well explained in this source.