Normandy

I assume you are flying open jaw into London and out of Paris which is perfect. Get Eurostar tickets as early as possible for decent prices. I assume you will have to come to Paris and then head for Normandy. I would pick up a car on arrival and drive to and in Normandy. This is absolutely not day trip from London travel.

I would stay on MSM -- it really is stunning at night when the buses have left -- We loved Honfleur, the cliffs of Etretat, the neolithic sites around Carnac and ST. Malo. We did a tour of the landing beaches. With a car and some planning, you could easily do the key spots on your own. I seem to recall big parking lots at the various sites. Maybe do Pont du Hoc, the American Cemetery and Omaha? Bayeux and the Tapestry not to be missed.

You definitely cannot do Normandy as a day trip from London. Most wouldn’t even recommend it as a day trip from Paris. Take the train from London to Paris, then to either Bayeux (where I recommend staying), or to Caen (which has more rental car facilities). If you rent the car in Caen, it’s only 30 minutes drive to Bayeux. Spend the next day (or two) touring the beaches. We really liked the 360 Cinema in Arromanche as a starting point. We also did the Overlord museum, Airborne museum in Mere St. Eglise, and Utah Beach museum. We also went to Pont du Hoc, Omaha Beach and the German batteries at Longues sur Mere. We spent two days touring these areas in Normandy.

We didn’t go to MSM, but if we did, we would stay overnight there. It seems like one of those places that is completely different once the day trippers leave. MSM is 1.5 hours from Bayeux, so it could be a day trip from there.

This is the apartment we stayed at in Bayeux in Nov. 2019. Great place, great location. https://abnb.me/6Pgum9MfZsb

As much as we enjoyed Bayeux, my boys aged 16 & 13 were bored stiff by the tapestry.

Thinking out of the box here.

Travelling from London to Normandy is surprisingly time-consuming due to the need to take the train to Paris (2h15min), change trains between stations in Paris (1hr is enough but allow 1h30 as a first-timer), then take the train to Caen (2h15 min) if you are interested in the D-Day beaches area.
There used to be flights from Southend to Caen, but not anymore, sadly.

However, Easyjet is currently flying 2x/week from London Gatwick to Rennes, in the late PM.
if they still do so next May (too early to know!), it would be very helpful (as much as I hate to recommend a short flight...): you could fly to Rennes, rent a car there, spend the night close to Mont Saint Michel, visit it early the next day and keep going towards the D-Day beaches.

If not, well, the train route through Paris works, annoying as it is.

By the way, D-Day beaches are especially busy around June 6th. Unless you want to be there for the anniversary, try to arrange your schedule in such a way that you miss it!

If you stay at Mont St Michel, you might enjoy walking out on the mud flats with a guide or tour. My kids are a little younger and enjoyed having a nature break. And even when the island is busy, the mudflats have a lot of space.
Also check when the highest, most dramatic tides occur; I think it is near the full moon.

https://community.ricksteves.com/travel-forum/general-europe/london-normandy-paris-help-with-itinerary-transport-options

Thank you all for your advice. Sorry for my delayed reply. Covid has stolen my zest for vacation planning, but the more I read, the more excited I'm getting and I think I'm already behind! So far I'm thinking train to Paris (from London) then train to Caen or Bayeux to rent a car. One, maybe silly question, do I take this rental car to MSM overnight? can I do this easily? We will likely take a train back to Paris or depending on time maybe take the car back and leave it at/near CDG in the evening before our flight home the next day.

Another fervent vote here to spend the night on MSM itself. It will seem expensive for the room you get, and the hotel probably won't leave you impressed by itself.  None of that matters.  The key is being on the Mont when the crowds are gone, and for that, the over-priced basic hotel is a worthwhile splurge for one night (you are paying for location location location). After you get home, you won't remember the price you paid, but you will remember the experience of being there at night, and in the early morning, when the crowds are all gone. You will want to book your hotel on MSM soon, there are only a few, they are small, and they book up far in advance.

To your question: yes, you can rent a car pretty much anywhere and driving in Normandy is quite easy.

Yes, take the train to Paris from London. Of course, plan to spend a few days in Paris, too (you're going to pass through there anyway, it would be a crime to miss it). After Paris take the train to someplace in Normandy, pick up the car there (you do not want to drive in Paris). Check with your hotel about the car logistics (there's no parking on the Mont, I think even for hotel guests, you will probably need to use one of the lots on the mainland and take a shuttle bus; your hotel will have all the details you need).

As suggested above, given the timing of your trip, be cognizant of the annual extra burst of tourist activity around that week in June. Places will book up very far in advance. That cuts both ways: there will be extra ceremonies to commemorate the D-Day landings (fascinating if you are there) but expect more crowds and more places that are booked up around that time. So figure out your plans accordingly.

Normandy is lovely, one of the most beautiful parts of a beautiful country. You and your family will love it.

Edit to add: Everything that David said.

If you spend the night on MSM itself, most places have an access code you can enter to park at the closest parking lot, from where you will board the free shuttle bus (no one can park on the island itself, but parking close and/or cheap is helpful). If you stay off island, I would just inquire as to parking at your hotel or B&B. The drive itself from Bayeux to MSM is about an hour and a half if you don't stop for sights. But the wonderful town of Avranches is a great stop for sights and a meal—although quite close to MSM. Other than that, there are some historic WWII towns at which you could stop, but otherwise most of the drive is just a super-slab. As far as a rental car drop-off charges, it depends on your rental car company. Some charge, some do not.