We all should know that
Prepositions are usually followed by a Noun (That's why it is called preposition - Pre + Position. The word whose position is just before a Noun.)
Now consider two sentences
- This is my pen and that is your pen. (Right sentence)
but if we don't want repetition, so we avoid "pen"
The sentence gets short
This is my pen and that is yours.
NOTE: "yours" includes a noun, which is a pen
but
"your" DOESN'T include a noun
similarly
"Hers" includes a noun
"HER" DOESN'T include a noun
now as told earlier, propositions are placed before a noun.
So, There is a friend of...? My/mine??
"My" doesn't include a noun
"Mine" includes a noun.
After a preposition, there has to be a noun or a noun case (possessive pronoun)