In the preface to his translation of the Chida's diary, Benjamin Cymerman writes as follows:
Ha'im Yosef David Azulai was born in Jerusalem in 5484 (1724), or
thereabouts. The exact date is not certain and some historians place
his birth as late as 1727. He is descended from a long rabbinic line
bearing the name Azulai. The origin of this name is not certain; it
has been pointed out that its Hebrew letters are formed from the
initial letters of the first five words of verse 7 in Leviticus chap.
21 but this is certainly accidental (and certainly irrelevant). The
family had fled from Spain seven generations earlier, and there may be
some unknown connexion with the Spanish azul meaning 'blue'.
In the Jewish Encyclopedia article "AZULAI, AZULAY", Louis Ginzberg, Gotthard Deutsch, and A. Porter discuss this as well:
Ḥayyim Joseph David Azulai (see No. 4) derives the family name from
the initials of the Hebrew words אשה זנה וחללה לא יקחו ("They shall
not take a woman that is a harlot, or profane," Lev. xxi. 7). This
derivation, however, is not at all probable; and it is to be presumed
that the name refers to a locality in Morocco or in Spain.
This is further discussed in the Jewish Moroccan Archive:
Azoulay is represented here by Moroccan and Palestinian creators. The
name is common among Jews in North Africa. It is associated with a
variety of meanings: Izil = good in North African Berber, the good
one; Tazoulaït = a tribe in Morocco; Azul = blue in Spanish (Azuelos),
the one with blue eyes and most interestingly an abbreviation of ‘isha
zona vahalala lo yikhu‘ (they shall not wed a prostitute or engaged
woman) which implies that Azoulay were priets in an ancient time.
The name may be an Arab/Spanish transformation of Oziel into AzOullah
= God beloved. The name may be related to ‘Az’ = ‘strong’ or ‘strength.’ Aziza may be an Aramean transformation of the same name.
The names: A’z, O’z, O’ziel are modern Hebrew version used in