All of the variations of the word are easily traced back to the origin: Trebium. This, instead of the root, Treb, appeared in many ancient Italic place names, as discovered by archeologists, like Trebula, found in Campania, Tribia, a tributary of the Po, and Treba near the source of the Aniene. The root word, used in so many locations, was used in much the same way as the word “Casale”, a vague term that connects to hamlet, farm, or farmhouse. All of these places that begin with the word “Treb”, has nothing to do with the word we are discussing. These are all far from the area of the Trevi Fountain, and where the aqua Vergine was.
The exact opposite is actually the truth. In the middle ages, as well as in ancient Rome, several “Casali” or the tiny hamlets that existed in the same area between Rome and Tivoli. There name did share the same root: “Trebaria land on the border with Tivoli” was given to the Anicia people in the 6th century; Pope Paschal was taken prisoner near “castellum Trebicum”, also situated between Tivoli and Rome; a huge farm, named in two notarised deeds of 1389 as “Lo Treio Mandarino” and in Latin as “Tribium Mannarinum” existed outside the walls of San Lorenzo on the Via Tiburtina”.
Also, in 1511, several parcels of land was given to Pope Eugene III from Oddo colonna, in exchange for the “totum castrum Trebani”, which happened to be within the exact same location where the springs flow that supply the Trevi with water.