Old English þin, possessive pronoun (originally genitive of þu "thou"), from Proto-Germanic *thinaz (source also of Old Frisian, Old Saxon thin, Middle Dutch dijn, Old High German din, German dein, Old Norse þin), from PIE *t(w)eino-, suffixed form of second person singular pronominal base *tu-. A brief history of the second person pronoun in English can be found here; see also thou.
Old English min "mine, my," (pronoun and adjective), from Proto-Germanic *minaz (source also of Old Frisian, Old Saxon Old High German min, Middle Dutch, Dutch mijn, German mein, Old Norse minn, Gothic meins "my, mine"), from the base of me.
As an adjective, "belonging to me," preceding its noun (which may be omitted), it was superseded from 13c. by my when the noun is expressed. As a noun, "my people, my family," from Old English. In this heart of mine, no fault of mine, etc., the form is a double genitive.