So I was pondering around various ideas before when my family and I finally settled on doing one of two vacations. We're doing around 10 days, not including travel time, and need help on which to choose. The trip is going to be with my husband, 15 year old and 8 year old daughters in the early or late summer. First, we would do 4 days in Paris, 3 days in London, and 2 days in Dublin. The second option is Italy. We would do 4 days in Rome, 3 days in Venice, and 2 days in Florence. I know that with the first option it is not supposed to be about the amount of countries, but the experience, but they are all conveniently close and easy to get to. And with Italy, I know it's supposed to be treacherously hot, but we can take it, especially to visit somewhere like that! We all think that either trip would be absolutely amazing, so we can't decide and definitely need other people's input on which to choose. And please don't say it depends on our interests, since all of us can easily find at least one activity they like on each vacation. We just want to know what others prefer before making a final decision.
You have 10 days. Does that mean 9 nights? That sounds like a long time, but in reality it's not. Also consider the time and expense of moving 4 people from place to place. I know you said you don't mind the heat, but keep in mind that there isn't much A/C in Europe.
With 9 nights, I would split the time between London and Paris. Fly into London (it's less expensive to fly in than to fly out to due airline taxes/fees), spend 4 nights, take the Eurostar to Paris and spend 5 nights. Fly home from Paris. I would suggest apartments, as you will have more space than small hotel rooms.
I think Andrea has the right idea. You are probably not a very nimble group and will get more out of less, if you take my meaning. And there is so much to see and do in Paris and London.
That said, I do think an Italian alternative is quite feasible. I'd suggest flying into Venice and spending 2 nights, not 3, then Florence > Rome.
Choose one city pair: London + Dublin or Paris + London. Tip the balance of more time to London either way.
Book apartments and plan on a day trip or two, if you'd like to visit more.
In another, similar, thread, I think you said that Boston was your nearest airport?
If you decide to do the trip in late summer (does that mean August?) I would do the Paris/UK trip. Hot in Europe always feels hotter than here. And the places that do have air conditioning don't keep it as cool as we usually do. And the big tourist sites will remain open but some restaurants and shops close in August in Rome because people go North on vacation. But in early summer (June?) the heat is tolerable, in my opinion. And I am a Midwesterner.
Italy is my favorite country and Rome my favorite city. So that would be my choice. But London and Paris are wonderful too. Dublin is my least favorite of the cities you listed. Is there a time period in history that is more interesting to you? A particular piece of art that you want to see more than the others? Are you catholic and want to go to a papal audience? You mentioned that you would all find something you like on each vacation but what do you look forward to most? If you close your eyes and picture your vacation, what are you all doing?
Maybe to help you decide, start looking at hotels and airfare. You might come across a deal that helps you make your choice. London, Paris, and Rome are Europe's most expensive cities. For a family of four, apartments are a good way to save money but you often have to rent for a full week. I think of the three cities, there are more options for cheaper hotels in good locations in Rome. But that might be because I know the city better.
As all 4 destinations are fabulous, with plenty to see and do in any one of them over 10 days, and if weather is any kind of a tie-breaker (and it's been mentioned a lot here), consider that summer's a better time than fall or winter to see Ireland. It WILL rain there no matter what time of year you go, and you can dress appropriately, but cold and wet weather is one thing, and (relatively) warm and wet weather is another. Assuming your family is able to make a return trip, and if you would have options in the future about exactly what time of year to make a future trip, perhaps Italy could be saved for a time when excessive heat/humidity might be less of a factor. We've been in Italy (north and south) in June, July, December, and January, and likely won't be going back again soon in the summer if we have other timing options. Happy travels!