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""Friend of mine". You could say "House of mine""House of mine", or "School of mine""School of mine", about places where you spent dearmemorable moments of your life, or "child of mine""child of mine" about your dearest baby. Saying "Lady of mine" is a fancy way of introducing your girlfriend or wife.
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""This is a fork of mine" with tears in your eyes - that's the right context for this usage.