of mine is an old construct that means not only possession but also emotional attachment.
It's currently rarely used outside of "Friend of mine". You could say "House of mine", or "School of mine", about places where you spent dear moments of your life, or "child of mine" about your dearest baby. If you use it in context of common items, it will have comedic effect. Imagine yourself cradling a fork to your heart tenderly, and whispering "This is a fork of mine" with tears in your eyes - that's the right context for this usage.