The latin name was Mediolanum. Apparently it comes from Celtic Medhelan, in the middle of the plain, and it should be a common name also on the northern side of the Alps (any French person can confirm?). I googled a bit to find when the name switched from Latin to Italian with no success. You could check https://en.wikipedia.org/wiki/History_of_Milan but I suggest to give a look at the Italian version of the page with more maps, if you're interested in that sort of things.

> I googled a bit to find when the name switched from Latin to Italian with no success.

That isn't the kind of thing that happens at a particular point in time. It's a process that takes hundreds of years.

I was thinking about the name in Lombard - Milan - which is the same name used in some other European languages. But I don’t know when Italian became the main language in the region.

Just like the name in the local language is also the one used across Europe for Turin - by the way.