[Sanchome]

I was thinking more about it, and I think it's true that "my friend" sounds more personal in some circumstances.

If someone is upset - very emotional - I think they would often choose "my friend" instead of the other two options.

So, if someone is mad and they start yelling, they might say:

Please, just be my friend right now!

Or if someone is really lonely they might say:

I need someone to be my friend!

If someone close to them died or moved away, someone might say:

I miss my friend so much!

In all of those cases, I think that "my friend" is much better than "a friend of mine" or "one of my friends".