The correct option is
A She had arranged to do something else.
The given situation states a past event 'invited'.
Option A: 'Had arranged' is in past perfect tense. It states an action that completed in the past.
The past perfect tense indicates that an action completed 'arranged to do something else' before another action took place 'invited Rachel to the party'.
Hence this option is relevant in meaning and A is correct.
Option B: 'Has arranged' is in present perfect tense. It states an action that has just happened. But the situation has already happened in the past.
Option C: 'Arranged' is a simple past tense verb. It states an action that has happened, but does not indicate when. The past perfect 'had arranged' indicates that the action happened before 'Rachel was invited'.
Option D: 'Has been arranging' is in the present perfect continuous tense.
It states an action that is continuing at present, as opposed to the past situation.
Hence options B, C and D are incorrect.