Guale
That was the Spanish name for the land north of Florida, what's now Georgia. Comes from the name of the local native american group there.
I thought that was just one of the many "provinces" Spain had described there. There were several, and Guale was just the main one in coastal Georgia and not Carolina or the Gulf Coast. Not that it couldn't be expanded to become the name for everything in the region.
Might work: Nuova Italia or Nuova Sicilia would probably be more popular. Cities and towns might be named after the ones in the old country, with a "Nuova" (New) in front: Nuova Napoli, Nuova Palermo and so on; in addition I suppose there will be a lot of them named after saints popular in the south of Italy (san Gennaro, Santa Rosalia, San Nicola) or Catholic feast-days (same as it happened in Spanish America). State-equivalent entities might again be named after the regions of the old country (Nuova Calabria for example) or after the names of princely or ducal families (for example, the land granted to the Orsini family would be called Orsinia)
If it broke away and made it's own country, I bet it wouldn't end up with a name like Nuova Italia or Nuova Sicilia considering how there are zero countries in the Western Hemisphere with names like that and only one independent state historically (New Granada, nowadays Colombia).
Also, as we can see throughout Latin America, a lot of the times they didn't even bother to put "New" in front of the name, just added to name, like "Cartagena de Indias" or "Córdoba de la Nueva Andalucía". Seems like Barcelona in Venezuela was originally called "Nueva Barcelona", though, but that seems to have gotten dropped over the years.
And going by the number of indigenous toponyms throughout the Americas (especially in Latin Americas), a lot of native names will survive too, including possibly in places where they didn't today (or vice versa).