Hotel in Nice with airport shuttle

I can't recommend a specific hotel, but booking.com may be of help. Airport shuttle is one of the filters you can use to search for accommodations in Nice. But any hotel should be able to book a taxi in advance for an early morning pickup.

Airport shuttles seem to be less frequently offered in Europe, probably because public transportation to the airport is nearly always available. Even if you find a hotel with a shuttle, it might not start early enough for you.

Tram Line 2 runs out to the airport. The first departure is at 4:40 AM. The trip from one end of the line to the other takes 35 minutes. From your hotel it would probably be a shorter trip. The departure times are shown here:

https://ftp.lignesdazur.com/ligne_2.pdf (Scroll down for trips to the airport.)

Travel times are shown here:

https://www.lignesdazur.com/fr/schedules?context=ScheduleLine&line=115

The airport is only about five miles from Nice area hotels. However for our last 6:30 am flight out of Nice we had a Uber/taxi hiccup and decided this time to stay at the airport.
The Novotel Nice Arenas Aeroport stated “Our free hotel shuttle runs 24/7 and leaves the hotel every 30 minutes on the hour and half past. You only need to come down 10 minutes before and ask for it at reception. There is no charge for the guests for the shuttle.”
If you join the free Accor group you get a discounted member rate.
We have not stayed there yet. BC gives it 8.4 with over 3k reviews so I’m expecting clean and convenient not too much more.

Because Taxis and Uber/Bolt generally work fine, or a private transfer, you may not find as many shuttles (as people noted).

If you do stay at the Airport, which typically isn't necessary (and not for more than a night), the Sheraton, Crowne Plaza and a couple of others are right near the Grand Arenas tram stop, which starts early and drops in front of the terminal. With not much luggage, you can even easily to Terminal 1 from those hotels (the Sheraton in particular).

I'm thinking about the French Riviera as well, from my research it looks like at least over half of these hotels do airport shuttles from the hotel to the airport. I dont think that will be an issue at all.

There's a company called a Friend in France that does airport pickups and dropoffs in Nice
CAR 1-3 ppl https://share.google/LVmoiEnHU1NkEGiMH
I used them back in 2019 and had a good experience - but this was before the tramway went to the airport.