Non-Schengen country - Will this extend 90 day visa?

Will he have a student visa for Italy?

You can try using the Schengen visa calculator to see

https://www.visa-calculator.com/

This program did not provide student visas. It was designed to be 90 days only.

The answer is yes, any days he spends outside of Italy in Ireland will add to his ability to stay after his program ends. I suggest he spend as much time as possible in the UK on spring break to save days. The rule is 90 total days in a 180 day period which starts the day he arrives in Italy.

either would work, as would any other non-Schengen country

but if he goes to the UK he will need an ETA

Any part of a day in the Schengen, means a day used, even if an hour or less. So in your original question, he is spending 5 days in Ireland, but since his arrival day, and his departure day will be partly in the Schengen zone, Yes, he only gains three days in the Schengen area.

Ireland and the UK are easy options if he has other time he can spend out, otherwise he might look to the Balkans, Montenegro, Albania, both close, even Bosnia and Serbia would be options.

You are correct. It is always a look back every day. Where has he been been for the past 180 days. You have only 90 days in the prior 180 so counting carefully is important and you appear to be very aware of the situation. Now if he does do a few days out of the schengen zone, he needs to be sure of passport stamps or other documentation since it is so tight. Good luck

Your son is not in the Schengen Area for the purpose of tourism and consequently those rules don't apply to him. If he does not have the appropriate Italian student visa, then it really won't matter if he breaks a set of rules that don't apply to him! I'd strongly advise your son to legalise his situation in Italy long before then.

Every year we have young people falling into this trap and the outcome is always the same deportation and a lengthy ban from reentry.

Your son is not in the Schengen Area for the purpose of tourism and consequently those rules don't apply to him.

This is simply incorrect. The 90/180 Day Visa Waiver program covers many uses other than just tourism. Most business travelers travel under the visa waiver, visiting family, medical treatment, cultural exchanges, academic travel, studies, providing and receiving training, and some volunteer activities are allowed, among others.

In the case of the OPs Son, his program is under 90 Days, and is likely under the category of "Study Abroad" meaning he is attending a program at an Italian University, but technically not a student, since he is taking the class at his US University. But even then, short term study under 90 days is still allowed.

it really won't matter if he breaks a set of rules that don't apply to
him!

I dont know if Jim or Paul is correct. What I do believe strongly, is that when son goes to leave Schengen after 92 days the guy stamping his passport at passort control isnt going to want to talk about son's education experience and possible exemption. That will have to be done with a lawyer before son decides to return to schengen in the future.

Back in 2018 I took an over-90-day trip to Europe. I had enough non-Schengen time embedded in the middle of the itinerary to keep myself legal; the trip began with Hungary, then Ukraine (non-Schengen), then Poland and other Schengen countries. When I departed from Czechia, my last Schengen country, on a flight to London, the exit Immigration officer's eyebrows went up when he found the original inbound Budapest stamp in my passport, because that date was about 3-1/2 months old. But everything was fine once I pointed out that I had spent over 3 weeks in Ukraine.

These days, I think they can just scan the passport and depend on their system to tell them whether that passport has spent over 90 days in the Schengen area. However, I'd suggest trying to get the passport stamped on the way in and out of the Schengen zone, retaining the airline/train/bus tickets used for those trips and having them readily accessible at the time of final exit from the Schengen area.

Yes, I made a passport control guy upset once fairly recently because he scanned my passport, got up, got assistance from another agent, gave me a "you criminal" look before I remembered and held out my residency card. I just figured the system would tell him but .... I got scolded and passed on. Now imagine the son trying to explain he is exempt with no card, even if it were true, but I doubt it.

From the US State Dept. I put certain parts in bold letters relevant to this thread:

Traveling in Europe

If you plan to travel in Europe, you need to know about the Schengen Borders Agreement, which allows you to move freely within a number of countries without border checks. Tourists, exchange students, and people visiting for business from certain countries, like the United States, can travel in the Schengen area for up to 90 days. The Schengen area includes most EU countries, except for Cyprus and Ireland. It also includes four non-EU countries: Iceland, Norway, Switzerland, and Liechtenstein.

Why don't you plan on meeting him in Ireland or the UK? It would be best if he got exited Schengen within 87 days. All you gotta do is have his flight cancelled with no new flight for several days and he cannot exit until then. Your friendly passport control officer may not have any sympathy for his plight. Nothing like a fine and being banned for 10 years!