8 Days — Paris to Normandy to Paris?

Assuming you have 8 days actually in France, and your travel days into and out of France are extra, yes its doable. You would want at least 2 nights in Normandy. If you want a car, may I suggest using the train between Paris and Caen; picking up and dropping off the car there. If the D Day beaches are your aim, stay in Bayeux.

Yes, I agree with CJean. However here's the math so you can decide for yourself if this is your travel style (I count nights rather than days when planning my travels).

-Nights 1 and 2 in Paris (you'll probably arrive around noon if flying from the east coast), so 1 1/2 days exploring Paris.
-Nights 3, and 4 in Normandy: get up early, train 2 1/2 hours to Bayeau, spend most of 2 days with a guide which will enable you to see the American beaches (assuming your interest is D-Day). If you want to see the Tapestry and take a walking tour of Bayeux, take a late afternoon train back to Paris, 2 1/2 hours. I mention a guide because the D-Day sites are spread out and navigating and finding parking at each one eats into sightseeing time.
-Nights 5, 6, and 7 in Paris: 2 more days in Paris, more if you skip the Tapestry and return to Paris in the morning. Fly home on the eighth day.

If you have one more night you can sleep in and catch a later train to Normandy, so nights 3,4, and 5 are in Normandy. And if you definitely want to drive you'd probably see more of Normandy if you spend three nights there. Normandy is a big region and there's lots to see. If you want to add Mount Saint-Michel that will cut into your time in Paris.

I would take the train to Bayeux the day you arrive and do all your nights in Paris afterwards, rather than moving from one hotel to another twice.

I would take the train to Bayeux the day you arrive and do all your nights in Paris afterwards, rather than moving from one hotel to another twice.

This is what we were originally thinking but it’s a 9 hour overnight flight to Paris for us so we’re thinking we will be tired and jet lagged and adding a train on after may be too much.

If you want to have a car to tour the D-Day sites, then my recommendation would be to take the train from Gare du Nord to Caen. Pick up your rental car there; stay in Bayeux; return the car to Caen and train back to Paris. Be sure to check the days and hours of operation for the car rental agencies. Many (if not all) are closed on Sunday, and they close for a lunch break. On our first visit to Normandy, we hired a private guide for one half day upon arrival in order to get an overview of some sites. We then used our rental card to tour at our leisure on the following two full days.

What TravelBug79 said.