Similarly, mezzogiorno in Italian is also used to refer to the southern part of the country. (The explanation is obvious by looking at a sundial).
I gather that some 60 - 70 years ago (no idea what has happened to it since), there was in the "heel" of the Italian peninsula, a privately-owned railway 80km. or so, long, which was actually named the Ferrovie Mezzogiorno. Bryan Morgan in his The End Of The Line, speculating about this line -- which he had never in fact visited -- writes waggishly that "[the] name quite misleadingly haunts me with its hint of a single train which can only leave when the sun is at its exact zenith".
The English translation of this sense of "midi" is "meridional", an archaic word for south.
This thing of "meridional" or equivalent, with south-related connotations, would seem to occur in Italian too. Morgan's next sentence after his above "Mezzogiorno" quip, tells of (situation as at the mid-1950s) "the busy but narrow-gauge electric Strade Ferrate Secondarie Meridionali which run from Naples... to places like Pompeii, and to Sorrento..." .