To express a possessive you can say "my X" or you can say "X of mine". The two constructions mean pretty much the same thing. No, "of mine" is not archaic, it's routinely used. But it is far less common than the "my X" form.
Bob is my friend.
Bob is a friend of mine.
Both mean essentially the same thing.
It's not limited to "friend", though now that you mention it, that may be the most common word used there.
The red one is my chair.
The red one is a chair of mine.
That said, "X of mine" is mostly used when you want to indicate that something is one of many that you own or are associated with. If, say, you have many cars, you might point to one and say, "That is a car of mine." But if you only have one, you normally say, "That is my car." (If you have many, you could also say, "That is one of my cars.")
That is, it's almost always "an X of mine", not "the X of mine". I don't think I've ever heard someone say, "That is the car of mine" or "She is the mother of mine"; it's always "That is my car", "She is my mother", etc. "The X of mine" would be grammatically correct, but no one says it.
Arguably it should be perfectly good to say "of me", there's really no need for the word "mine". But that's just not what English-speakers say, it's always "of mine". The only time I can think of when we say "of me" is when "of" is not indicating possession but rather is being used to mean "about", like "That's the story of me." (Even then, you'd be more likely to say "of my life" or "of my job" or whatever.)
(There is the phrase, "You're not the boss of me", but I think that's deliberately incorrect for effect. If I was simply stating the fact that you are not my boss because I have been assigned to a different department or some such, I would say "You're not my boss." "You're not the boss of me" is something you yell when someone's trying to order you around and you do not recognize them as having any authority, like when your big brother tells you to shut up.)
Like any construct that is not normally used but that is grammatically correct, people sometimes say "X of mine" for emphasis or poetic effect. Like I wouldn't be surprised to see "oh wife of mine" in a love poem or a Valentine's card, but no one says that in normal conversation, it would always be "my wife".