The authors agree that the surname was formed from the personal name of Germanic origin Haimirik or Heimrich, composed of haimi/heim ("homeland") and rikia/erich ("powerful, lord, prince, king"), meaning "lord of the homeland" or "sovereign". The name was Latinised as Henricus and Italianised as Enrico.

E. De Felice specifies that the name, in the Ostrogothic and Frankish tradition form Haimeric or Haimirich, continues in the Italian Americo. The Old High German form Haimrich, Hainrich and later Heinrich, spread by the imperial chanceries in the Latinisation Henricus, is at the base of the Enrico type that became established in Italy between the 10th and 13th centuries through the prestige of the German kings and emperors of this name.

De Felice notes that Arrigo is a mainly Tuscan variant, and that Enriquez —with the forms adapted to Italian phonetics Enriques and Erriquez— is a direct loan from the corresponding Spanish surname Enríquez.

De Felice adds that the surname is widespread in the Arrigo type from Tuscany to northwestern Italy; in the Enrico and Errico type in the peninsular south —especially, with Errichiello or Arrichiello, in the Naples area—; and in the Endrigo, Endrici, Endrizzi type mainly in Trentino.

E. Rossoni agrees on the Germanic derivation and adds that in the 13th century the noble Florentine family of the Arrighi was at the origin of the dispute between Guelphs and Ghibellines. In the 14th century the Arrighi appear among the well-off families of Lucignano (Arezzo area). In the 16th they occupy prominent positions in San Miniato (Pisa area). He also highlights, in the 17th, the wood sculptor Alessandro Arrighi, and in the 18th the architect Antonio Arrighi, both from the Cremona area.