(This is too long for a comment, but is not complete answer)
Surname origin is an interesting topic. Most cultures use the same basic inputs (geographical, occupational, personal and descriptive), but language, history and neighboring influences make a more individual mix.
The following book includes Apap, but the excerpts at GoogleBooks are too limited to evaluate:
The surnames of the Maltese Islands: an etymological dictionary (Mario Cassar, 2003)
It may be available through a local library.
The article Place-names and Personal Nomenclature of Gozo in Oriental Studies (1980), most of which can be read at GoogleBooks, implies that Apap is a surname imported to the island after the sack of 1551, but does not specify an origin. The article does discuss surname patterns that may be applicable to your name of interest.
When analyzing a name for possible origins, one must consider syllable and sound shifts (due to time and changing languages). A-pap or Ap-ap. P or B or V. A or O. Knowledge of the Maltese language would help tease out which trends really occurred. (FYI Ab(a) is Arabic for father)