On my way to Copenhagen but what to do with 3 extra days in France or Amsterdam

Picking up the car in France and dropping it off in the Netherlands will probably incur a very, very high international surcharge.

Last year I spent time in both Arras and Amiens after visiting Lille, then I went on to spend over two months in Belgium and the Netherlands. The possibilities are nearly endless.

What sort of sightseeing do you like to do?

I think most people would find more to keep them in busy in Amsterdam than in Lille, though I spent more than 3 nights in Lille myself (compared to 2 weeks in Amsterdam).

Do not rent a car in one country and return it in another country unless you are prepared to pay 500-1000 euros extra. Dropping a car off in a different country entails lots of complications for moving the car to its country of registration. (It's not like different states in the US.) Stick with your train plans.
If you have extra time, Paris is always a good place to spend it. However, adding time onto Lille and/or Amsterdam sounds good also. Perhaps you could list the things you'd like to see and do in those two locations, and maybe look at day trips from them as well. I am sure you can fill in an extra day in Lille and a couple of extra days in Amsterdam.
Do be sure to check out what needs reserving ahead of time. I read that the Anne Frank house is tough getting tickets for. Visit a working windmill home, check out Edam, Volendam, the Van Gogh Museum, the Rijks Museum, ride a boat on the canals, join some strangers at a large cafe table for ristafel (sp?). Visit a tulip farm. The museums may also need pre-booking.
Stop for a day (or overnight) between Lille and Amsterdam at Bruges, or day trip there from Lille ( 1 1/2 hrs by train)
I don't have Lille suggestions as I haven't been there. Put Lille in the search bar above and filter for "forum" and "one year or less". You should find some info there, or see what you find on Tripadvisor. And check out the tourist bureau info online for Lille.
Enjoy your trip!

Your itinerary is crying out for a couple of days in Belgium. Antwerp, Brussels, Ghent, Bruges are the big 4. You could stay in one and commute to the others, and lose no time on the way to Amsterdam.

Lavandula

Thanks for letting me know about car rentals between countries. Since our stay in Lille has agreed to extend for a couple of the extra days, I'm thinking we might try renting a car from the Lille train station and visit Belgium as suggested or other nearby places as day trips. Perhaps I'll try to add one day to Amsterdam and do some side trips while there and rent a car as needed.

Thanks for asking, We like to see museums and sights, we like to go on 2-3 mile hikes, and listen to music whenever possible.

I agree with Lavandula; go to Belgium. You can easily take a train from Lille to a city like Brugge, Ghent or Brussels. From one of these cities you can then take the train to other Belgian cities. And when it’s time to go to the Netherlands, the train is again the easiest form of transportation.