This surname derives from
a Christian name, Gaddo, short for Gherardo, used mainly in Tuscany,
which was formed by syncopation and assimilation of internal phenomena
(Gherardo > Gardo > Galdo > Gaddo), but since the surname
is documented also in Northern Italy, the base could be also a
Germanic diminutive Gado (from Gad = fellow). A Gado is
documented in 1157 at Genoa in the "cartolare"
by the notary G. Scriba, while a Gaddo is mentioned in a document
written in Pisa in 1329: "...hereus de Gaddo de Visinyano,
... notari..." (link) and
in the "Cronica" of the Saint Catherine monastery
in Pisa, compiled between 1390 and 1411, several friars called
Gaddo are mentioned: "Frater Gaddus de Verchionibus"
Gaddus de comitibus de Donoratico" (IV,
Gaddus, de famosis conventus nostri conversis" (VI,
Gaddus de Sancto Frediano" (VI,
Gaddus de Curtibus" (VII,
152) and finally
"Frater Gadduccius de Sismundis" (V,
Dante in his "Divine Comedy" (Inferno, Canto 33rd, 1-90), talked about a Gaddo, a really existed person, who was the son of the count Ugolino della Gherardesca. Gaddo, in real life like in the book, starved to death in 1289 with his father, his brother and his nephews, all locked in the "Muda tower" also called "hunger tower" in Pisa.
Another renowned artist bearing this name was Gaddo di Zanobi known as Gaddo Gaddi (12391312), who was a Florentine painter as well as his son Taddeo Gaddi (12901366), a pupil of Giotto, and his sons, Giovanni Gaddi, Agnolo Gaddi (13501396) and Niccolò Gaddi.
It is anyway also possible that the surname Gaddini derives from the name of a place: in the Diplomatic Codex of Middle Age Lombardy in an act of May 1123 it can be read: "...predicta vinea iacet ad locum ubi dicitur in Gaddo, est ei a mane Sancti Ambrosii..." (link).
On a website (link) a Jacopo Gaddini can be found, husband of Giovanna, from the Florentine family Ferrucci, who died of plague in 1363.
According to some source the Gaddini came from Romagna, later moved to Rome, then to Foggia (Apulia).
In ages not clearly definable Guglielmo was the prefect of Monza and doctor at law, while Giovanni, an Aurate Knight, was ambassador to Charles V.
In the Middle Age some Gaddini were bishops and later Guglielmo was the Bishop of Parma and Guastalla in 1642, and in the Eighteenth Century Giuseppe was the Bishop of Spoleto.
On the House of names website (link) a (dubious) Gaddini Coat of Arms can be found (image).
According to the Italian telephone directories of 1996, the 60% of Gaddini dwells in Tuscany, mainly in Lucca province (53%); 15,7% are in Latium (almost all in Rome) and 14% in Liguria (almost all in Imperia province).
On a Mormon website (link) I found two Gaddini, both 19th century women, one of which native of Lucca province.
From the list of the immigrants in transit between 1892 and 1924 at Ellis Island (New York), 61 Gaddini are reported, of which 42 from the commune of Capannori (26 from Massa Macinaia, 10 from Capannori, 4 from Marlia and 2 from Badia Cantignano) and 8 from the city of Lucca.
A map of Italy with the main cities and regions quoted in the text can be consulted here, and my Gaddini ancestors can be found in my genealogical tree (link).