My family of four spent last Christmas in Paris and had a WONDERFUL time. We rented an apartment from Paris Best Lodge...late last year, Paris city government was cracking down on illegal vacation rentals (long, long story on that...go to a travel site such as Fodors and search their forums for more info). To be safe, instead of AirBnB, we went with an apartment rental agency with a really good reputation, because I felt that if the apartment we rented was taken off the market because of issues with the city government, the agency would be able to fit us into a similar space. We had no issues at all, and I don't know if Paris is still cracking down (there are *so many* vacation rentals), but it's something to be aware of.
We stayed for a week and rented a 'large for Paris' one bedroom apartment so that the four of us, two adults and two kids, could stretch out. Our apartment was the street behind Musee D'Orsay, right in the center of the things we most wanted to see. We might have been able to find hotel accommodations for less than we payed for the week, but the hotel would not have been as centrally located and would not have had as much room or a fully working kitchen! The price was worth it for the space, and for the ability to eat at our apartment.
Speaking of eating...a great way to save money is to buy at grocery stores and eat at your apartment or to pack picnics. We had maybe four take out meals (falafel in Marais, two meals from stalls in the Christmas market, and takeout Chinese) and various pastries while we were on the go, but other than the falafel, our food made in the apartment was better. The quality of foodstuffs in the markets in Paris was just amazing. We had baguettes and pastries and cheese and pate and ham and fruit...we baked a fantastic chicken for Christmas, bought pre-made, refrigerated soup to warm up at home (my oldest son still talks about this soup), and of course had lots of chocolate. OH, and very good, very inexpensive wine. And probably the best yogurt I've ever had. A frozen pizza that my kids loved loved loved (it was really good). And a really, really, really good pre-made beef and carrot stew. And other things I am forgetting

.
Groceries are not cheap in Paris compared to most places in the U.S., but eating that way is certainly cheaper than eating out. Plus, the variety of quick to make food items (bread/cheese/pate) with really good pre-made things make food prep in the apartment easy peasy. Plus, plus, I love shopping at groceries outside of the U.S. and found great things to take home as gifts (soap, hand cream, chocolate, coffee, jars of pate, jars of herbs de provence, etc.). We didn't eat out because of our kids, but if we get to go again, I'll plan a couple of sit down meals and easily do the rest in our apartment.
Other notes:
1) Check out the museum card. If you plan on going to a lot of museums, the card could very well save you lots of money. However, you have to go to a LOT of museums to see the benefit. If you just want to hit a few large ones, you will do better paying entry fees at the museum. Caveat to that...the card can also save you time, because you can bypass long lines for same-day tickets and just go into the museum.
2) The Louvre is sometimes open late and is much less crowded during these hours. I don't have recent info, but years ago, DH and I spent Thanksgiving in Paris and visited the Louvre on a Friday night (IIRC). Not very crowded at all, which made it easier for us to see the sections we wanted to see in much less time.
3) This is not a cost saving measure, but we did a scavenger hunt at the Louvre with our kids through thatlou.com. It was absolutely a fabulous way to see much of the museum, hit highlights, but also see lesser known (but still spectacular) works of art. Very worthwhile for us. It was great for our kids, but I'd do it again with just DH!
4) This recent trip, we saved by walking a fair amount. We had two young kids, so walking was imperative to wearing them out...but walking from the Sewer Museum to the Military Museum back to our apartment behind D'Orsay was very doable. There are more interesting neighborhoods to stay in, but in being so central, we didn't have to rely on public transportation too much.
5) Parks are free, beautiful, and offer great people watching. We also like browsing in department stores...their kitchen and hardware areas are a treasure trove, lol. Just walking around side streets is free and offered fantastic window shopping. Outside of a handful of museums and monuments (Eiffel Tower, Louvre, Arch, Military Museum, Sewer Museum, Notre Dame/Gargoyles), we didn't spend money to see things. We just did a lot of playing and people watching in parks, walked through pedestrian areas and Christmas markets and department stores, and took night walks on the Seine to watch the Eiffel Tower glitter on the hour...and it was a great time.
Sorry for blathering, but Paris really was amazing!