Step 1: Recall the double genitive rule.
In English, we often say "a/an/this/that + noun + of + mine/yours/his/hers/ours/theirs" to mean one member of a group tied to a person, for example a cousin of hers, a habit of his.
Step 2: Match the sentence to the rule.
The given sentence is "She is a close friend of ____". The blank comes right after "of" and needs the absolute form of the possessive pronoun, not the adjective form (my) or the subject/object forms (I, me).
Step 3: Test the fit.
Placing mine gives "She is a close friend of mine," which matches the double genitive pattern exactly and sounds natural. The other three words break the pattern or the part of speech needed.
Final Answer:
The blank must take the absolute possessive pronoun that completes the of-phrase. $$ \boxed{\text{mine, option D}} $$