Como, Cinque Terre or Sicily in June?

Cinque Terre or Lake Como would probably work better with your planned destinations. Sicily would be more of a trek.

Lake Garda is supposed to be beautiful as well, and is fairly close to Venice, but I've never been

I think most would agree that Sicily is not a great fit here. Staying on the mainland will be a much more efficient use of time, and to make the effort worth the reward you would want to devote a good week to Sicily at a minimum.

I’d be kind of spoiled with three weeks here, and I think you could work both Lake Como and Cinque Terre into the itinerary if you wanted. If I had to choose, maybe Lake Como? If only because it is different (not Coastal) from where you have been/are going already.

I am not a fan of Cinque Terre so I would go for the Italian Lakes. I like Sicily but have spent my time there visiting Greek temple ruins, churches and museums - of which the Island has an abundance. I think the Lakes would be the best for a relaxation break.

Sorry, I'm going to vote for Sicily. Especially Ortigia. There is something to keep you relaxed and interested both day and night, with maybe some archeology if you want. I've been here a few times as my 'gotta relax' place. Ortigia is small, clean, safe and slow moving. Great for sunrises and sunsets. Wonderful food and local wine. Fly into Catania and either bus or car service. Don't rent a car - you can't drive it very much in Ortigia and parking is, shall we say, challenging. Hmmm, thanks. Now I have the yearn to get back there soon!
EDIT: I usually stay a week and leave very relaxed. 4 nights might do it for you.

Do you plan to have a rental car or relying on public transportation (train and/or buses) only?
Leaving the Naples/Sorrento/Amalfi aside, since you've visited before, these are some ideas for R&R in June.
From Rome: Ponziane islands (Ponza island easiest to reach). Fast hydrofoil or ferry from Anzio (take train from Rome to Anzio). 3 nights are adequate.
From Tuscany: Maremma area in southern Tuscany. A minimum of 3 nights is recommended. However a car is recommended for this area
From Tuscany: Cinque Terre. Take train from Florence (via Pisa). 3 nights is adequate.
From Venice: Lake Garda. Train to Peschiera del Garda from Venice. I prefer the north side of the lake (more majestic mountains), however the railway does not go to the north side. You can reach it by bus from either Peschiera or Verona (generally with a bus change in Garda town). 3-4 nights would be good. Malcesine is a popular and quaint town on the north side.
Lake Como takes longer to reach from Venice, compared to Lake Garda. The easiest way is to take a train to Varenna (one change at Milan Centrale). 3-4 nights is good.

Sicily is a very large island (3 times the size of the Big Island of Hawaii) and has a ton of sights of interest. You will need a couple of weeks to see it all, and at least a week to visit a portion of it (either the West or East side). If you don't have that kind of time don't bother. If you do, you need to fly to it. There are many flights from Rome to either Palermo on the West side or Catania on the East side.

Have you determined where you will be flying IN/OUT of?

Sicily needs a lot of time, most recommend at least 2 weeks.
It’s the outlier on your list and will require a lot of extra travel. I’d save it for its own trip ( or with Puglia)

I don’t think you’ll find CT a “relaxing” place in June.

Looking at your list I have visited all the destinations except for the Lakes. I’d be inclined to add a few days to Lake Como. This area sounds like time for relaxation! Would you be flying out of Milan? Another idea would be Lake Garda which is closer to Venice. Enjoy your 3 week journey with your daughter!

Sicily is usually quite hot by June, and as mentioned it is a long way from the core of your trip.

Everyone loves Lake Como, but for those seeking an area a bit less traveled I would also suggest Lake Garda...
https://www.visitgarda.com/en/garda_lake/

Or, if you are near Lake Como anyway, and want something different you might go a short distance into Switzerland to Lugano. If you want to make a major excursion you can do the Gotthard Panorama Express train through the Gotthard Pass and along Lake Lucerne by cruise boat to Lucerne - and then catch the fast train back to the start all in one day.

https://www.sbb.ch/en/leisure-holidays/trains-trips/rail-travel-specialtrains/panoramareisen/gotthard-panorama-express/timetable.html

I love Sicily and agree with not visiting Sicily in June. Your trip is too far north and the island is as hot as the hinges of Hades (have wanted to use that expression for a long time) in the summer. One of my favorite, least touristed areas is the Piedmont region and Turin. Great national parks in the area if you like hiking and nature. If you have a car, head to Courmayeur in the Italian Alps. Take the Skyway Monte Bianco to the top. If you feel adventurous, take the cable car across to Aiquille du Midi and then switch to cable car that’ll take you down to Chamonix. You can get a bus to return to Courmayeur.

Depends on when you will be travelling. We haven’t been to Sicily, but based on our trip October 2022, I would recommend Varenna on Lake Como for 3 nights, 2 full days and access by train. You can use the mid-Lake ferry to see the other towns. We did it by train from Bologna and after 3 nights went onto Stresa, Lake Maggiore.

Sicily needs at least two weeks. If you only go to one location make it Ortigia Is, Siracusa’s charming historic quarter. We spent a week. With travel there you should spend at least five nights.
My first choice for your trip would be Lake Como. We have spent up to two weeks there four times and love it. We stay in Bellagio which offers many dining choices and more frequent ferries. Stay at least four nights.
Lake Maggiore and Lake Garda are different from Como but still are beautiful lakes.

I agree with the others.... Sicily is probably not a great fit given the other places you'll be visiting. Also it deserves more than a few days.

Everywhere in Italy is probably going to be busy at that time of year, so you'll have to decide which place you most want to see. I like both the Cinque Terre (especially Monterosso) and also Lago di Como (Varenna). Depending on which order you'll be visiting the other locations, you should be able to fit either one into the Itinerary.

You might want to have a look at the RS Italy guide book to give you an idea of the activities and sight seeing at Como and the Cinque Terre. Whichever one you choose, I'm sure you'll have a great time.