[Sanchome]

Interesting question! It's hard to answer.

All of the phrases can be used the same in many sentences.

She's my friend.
She's a friend of mine.
She's one of my friends.

My friend told me something.
A friend of mine told me something.
One of my friends told me something.

I think those sentences are close to exactly the same meaning.

Maybe in this case, "My friend" is more approriate:

Do you want to be my friend?

It might sound a little less personal to say:

Do you want to be a friend of mine?
Do you want to be one of my friends?

I guess it is because "my friend" can be a little more personal. "My friend" might sound like a closer friend than "one of my friends".

It's a hard question!