Weather in France in October?

Wet and windy, warm and sunny, cold and grey, it all depends on the weather fronts coming in from the Atlantic. October can be a stormy month particularly if hurricane remnants sweep in.

Days (and nights) are more chilly in October but still there are some sunny days with temperatures between 15 and 20 C° and occasionally even higher. If Normandy means visiting the coast, means the possibility of strong winds from time to time combined with rain, so have a windbraker with you that will keep you dry too. Usual advice is dressing in layers.

There's a saying, "In Normandie, one experiences all seasons in one single day."
Be prepared for rain, don't bring an umbrella (useless in the wind).
No need for rain boots.

Weather is really changeable so it is important to have water proof footwear and layers. That said, we have spent several Octobers in Paris and toured Normandy in very late September and love this time of year. It tends to be cool enough for a jacket and sweater, but we have had relatively little rain and it has never been either hot or cold. It is precisely the weather we prefer for travel. You can see a late October walk we took here:
https://janettravels.wordpress.com/2011/11/01/stroll-on-a-sunny-oct-31-parc-citroen-to-the-eiffel-tower/
And here is what it was like in Normandy for us the last 5 nights of September a few years ago:
https://janettravels.wordpress.com/category/normandy/
When we were in Paris in October in 2015, it rained a lot only one day. This past fall, we had a couple of showers, but most days I was out and about in a lightweight travel shirt over a T-shirt. It is a beautiful time for Paris in our experience and we have spent 4 Octobers in Paris in recent years.

I also was in Paris, 12 - 18 Oct, in 2015. Cloudy, some misting, one day of rain off & on & some partly cloudy/sunny days. You can check Weather Underground, https://www.wunderground.com/weather/fr/paris/48.860001%2C2.350000, prior to your departure to get an idea of what it will be like. You can also access previous years through the Calendar Tab. For me the weather wasn't a major factor, all I cared about is that I was in Paris.

Enjoy your trip.