It´s not natural to say "friend of mine", one would say "a friend of mine" (um dos meus amigos).
The pattern here is to use the indefinite article ''a" plus "of mine", and it means that you have more than one. Same happens to other statements in this way:
a teacher of mine [um(a) dos meus(minhas) professores(as)]
a house of mine (uma das minhas casas)
a car of mine (um dos meus carros)
a friend of mine (um dos meus amigos)
In all of the given examples it means that (1) I have or (2) had more than one at a time.
Just to illustrate the point further...
Don´t use "This is a car of mine" for "This is my car'' it´s plain wrong. On the other had if you are talking about a list of past cars that you had or have, then you could.
Thus to talk about "this friend" you say "This is my friend/This is a friend."
You could say "This is a friend of mine.", though, if you want to emphasize that you don´t have just a friend, "this" friend is one of your circle of friends.