Specifically, why the 'gi' sound ? Sorry for writing in English, my Italian isn't good.
The name for Paris came from the ancient Latin term used by the Romans to call an earlier settlement, Lutetia Parisorum or "Lutetia of the Parisii".
The Parisii were Celtic Iron Age people who lived on the banks of the river Seine (in Latin, Sequana) in Gaul from the middle of the third century BC until the Roman era. With the Suessiones, the Parisii participated in the general rising of Vercingetorix against Julius Caesar in 52 BC.
Julius Caesar in De Bello gallico referred to that people, the Parisii:
(VII,57) Dum haec apud Caesarem geruntur, Labienus eo supplemento, quod nuper ex Italia venerat, relicto Agedinci, ut esset impedimentis praesidio, cum quattuor legionibus Lutetiam proficiscitur. Id est oppidum Parisiorum, quod positum est in insula fluminis Sequanae
(VII,58) Id est oppidum Senonum in insula Sequanae positum, ut paulo ante de Lutetia diximus.
As you can find in this etymology dictionary for Paris:
from Gallo-Latin Lutetia Parisorum (in Late Latin also Parisii), name of a fortified town of the Gaulish tribe of the Parisii, who had a capital there; literally "Parisian swamps" (see Lutetian).
The original denotation Lutetia transformed in Paris around 360 during the council about arianism heresy, but it seems that the definitive name Paris was given in 508, the franc king Clodoveo I fixed to Paris the name of the capital city.
You can also take a look to page 47 of this Grammatica delle due lingue italiana e latina written by Ferdinando Bellisomi around 1824, about the correlation to Latin and Italian names:
Ci ha per ultimo de' nomi, che usati nel solo numero plurale terminano generalmente per i, ae, a. Quei della prima desinenza, come Parisii Parigi, Puteoli Pozzuoli, sono maschili
So it seems that at the beginning of 19th century the Italian form was the current one.
I don't really know, but I suspect the double -ii is the reason:
- *Parisi /pa'risi/ -> /pa'rizi/ which would have been written 'Parisi' (the
*means this is NOT a known form, I give it here as an example of how things aren't)
- Parisii /pa'risii/ -> /pa'risji/ -> /pa'ris.Zi/ -> /pa'ridZi/ which is written 'Parigi'
Cf Perusia -> Perugia.
If indeed older sources have Parisi, it may have been an artificial classicising spelling. Such things are not unheard of.