Divine or of the Lord
Damonica is a modern American elaboration, likely created by combining the prefix 'Da-' with the name Monica. Monica is of North African or Latin origin, often associated with the Latin 'monere' meaning to advise or the Greek 'monos' meaning alone, though in this specific compound form, it is often interpreted through its phonetic similarity to Dominica, meaning 'of the Lord'.
Less common today
Damonica isn't in the latest US Top 1000. It last peaked at #4,304 in 1990.
US births per 5-year period, 1990βpresent (SSA data).