Gucci is not a made-up word or a place. It is a family name, and the founder's first name nearly matches it.
Guccio Gucci was born in Florence on 26 March 1881. His full name, according to Wikipedia, was Guccio Giovanbattista Giacinto Dario Maria Gucci. His father, Gabriello, was a leather craftsman from San Miniato, and his mother, Elena Santini, came from Lastra a Signa.
As a young man he went to London. Wikipedia's Gucci article dates his time at the Savoy Hotel from 1897, while the article on Guccio Gucci himself says 1899, so the exact year is unclear. The Savoy itself says he was a luggage porter, and the job let him watch what wealthy travelers wanted from their bags. Wikipedia adds that he worked for the luggage maker Franzi after the First World War and for four years with the Compagnie des Wagons-Lits, a European rail company.
In 1921 he opened a shop on Via della Vigna Nuova in Florence, selling imported leather luggage. The business was legally called Azienda Individuale Guccio Gucci, and it grew into Guccio Gucci S.p.A., which trades simply as Gucci. The brand name was never chosen from a list. It was the owner's name on the door.
What does the name mean? Ancestry's name-meaning page lists Gucci as an Italian name meaning "diminutive, small," and gives Gucin, Lucci and Guido as variants. That is a general surname note, not a history of this family, and it does not say when the name first appeared. Wikipedia does not explain the surname's meaning either.
The double-G symbol is often said to stand for the founder's initials, Guccio Gucci, drawn by his son Aldo. Brand histories disagree on when. Art de Vivre says the 1930s and NSS Magazine says the 1960s. Wikipedia does not mention initials. It records a double-G mark on canvas handbags during the Second World War and a double-G logo for belts and accessories in 1964.
Guccio died in Milan on 2 January 1953. Gucci opened its first store in the United States, on Fifth Avenue in New York, in November that year.