itinerary help please!

It’s ok
I think I’d change the order
Venice to CT
CT to Lucca
Lucca to Florence
Florence to Venice

This splits the 2 busy cities up and gets you to Venice quicker on that last day

Is there anyway you can change your departure flight to Florence?

Traveling back to Venice from Lucca to spend the last night there is counterproductive. Backtracking is expensive and unsettling.

Why so much time in the Cinque Terre and no time in the greatest city of the western world, Rome? Can you rearrange your trip to depart from Rome? This would allow a readjustment of your places to visit.

Lucca was great. Would I substitute it for Rome? Never. I've never been to the Cinque Terre, so cannot comment, but it's difficult to get to and from and may not be the most interesting site for your daughter.

Changing your path a little will save 1-2 hours train time.

Venice>Lucca> CT>Florence >Venice. Gives you a sea, city, sea itinerary.

What do you plan to do in the CT? I would describe the area as charming vistas of colorful villages nestled on the rocky slopes, but not spectacular. There's hiking the trails and tiny villages that can be explored in a couple hours each. The only sandy beach is the size of a large postcard in Monterosso, the other beaches are even smaller and pebbled. If you want to soak of culture with a good mix of history and art, spend your time in Bologna instead. It's a rail hub so you can easily day trip to a goodly number of charming towns. . . Modena, Ferrara, Ravenna (exquisite 5th-7th century Byzantine churches and stunning mosaics), Parma . . .

You are also shortchanging Venice. Add one more night. There's lots of art and history in a truly unique city. I like the idea of starting in Venice because it's the ideal place to get over jetlag and enjoy "being" in Italy. For your day/night, you could stay in Padua which isn't far from Marco Polo airport. It's about as easy to get there from Padua as from Venice.

I haven't been to Lucca.

Lucca is nice, but you can see it in a day, no need for three nights.
You didn't include Sienna, which I strongly recommend. We did it on a day trip from Florence.
Many love CT, it's good, but I like the Amalfi Coast better.

It's nice to see an itinerary that doesn't involve rushing around constantly, but I do think this one could be improved with some tweaking.

I like picturesque towns and villages and have been to the Cinque Terre several times (not in recent decades). I recommend thinking about what you want to do there. The towns are small and only minutes apart by train. I'd recommend making a boat trip along that length of coastline in addition to the usual moving around by train, so you'll get the water views. But unless you are interested in doing a lot of hiking (which you have not mentioned, so I'm guessing not), there's not much to do there for three whole days. It's impossible for me to predict what the tourist load will be like this June, but in normal years the villages are utterly swamped, which tends to get old (for me). I agree with the other respondents that some of those Cinque Terre nights would be better used elsewhere. There's really no comparison between Venice and the Cinque Terre in terms of things to do and see (for non-hikers, I mean). I'd definitely switch at least one night in Venice; you'll be arriving jetlagged and may not even be fully functional on June 13.

Lucca is also fairly small. With all there is to see in Italy, I wouldn't spend more than one night there.

What to add is a personal decision. Rome and Bologna (which has great day trips) are worth considering, though you'd be a bit short on time for Rome, I think. But it depends on what sort of experiences you'd like to have. You seem to be gravitating to smaller places, so I'll toss out a suggestion for one of the lakes, probably Como.

I do like Chani's idea of spending the last night in Padua rather than Venice if you must fly back from Venice. Padua is a very interesting university town with quite a lot of sightseeing possibilities. (If you want to see the Scrovegni Chapel, it needs to be prebooked.) Hotels in Padua are a lot less expensive than those in Venice, too.

We are flying into and out of Rome this summer, despite not spending any time at all in Rome, because our airline tickets were several hundred dollars cheaper that way, so I feel your pain about having to backtrack into and out of the same city. The good news is that (as others have already mentioned), there is generally quick and easy train service between large cities, so if the tickets in and out of Venice save you money, I wouldn't worry about trying to rearrange your flights.

What I would do is drop all your nights in Lucca and allocate them elsewhere. If you plan to get an Airbnb, I'd add all the Lucca nights to Florence. We usually find deals of 10% off the total cost when staying 7 nights vs 6 nights or fewer. From Florence, you can easily do day trips to Lucca and Sienna or other Tuscan cities.

If you’re not hiking or spending a day at the beach in the Cinque Terre you only need two nights. With four nights in Florence, I’d take a direct bus to Siena for the day (1h 15m).
I agree with Gerri about changing the order of your itinerary and going to Lucca from Venice (4h 15m) that requires a transfer. Taking a train from Venice to the Cinque Terre takes (6h 30m).
From Lucca you can take a train to Monterosso (1h 30m) and sleep there. If you stay at another spot on the Cinque Terre, you’ll need to make an additional connection. The same if departing from Venice.
When taking regional trains that require transfers you want to pack light because some regional stations have stairs you must climb to reach your track. It’s possible that there’s an elevator but good luck finding it.