Early morning flight out of Lyon - where to stay?

That's an early flight. Too bad you scheduled it.
I would stay in the area around Gare de Lyon Part-Dieu.
As to whether to take a taxi or the Rhônexpress (tram from Part-Dieu to the airport) depends on how many are in your group.
One-way tickets purchased over the internet for Rhônexpress to the airport are about 15€. Taxi fare is probably on the order of 50 or 60€.
In other words, if there are fewer than 4 in your group it's probably a better deal to take the tram than a taxi. The first Rhônexpress tram departs Part-Dieu at 0425.
The area around Part-Dieu is seedy, but relatively safe.
I'm not a traveler, though. I'm a former resident.

I was going to suggest the town where we're spending our last night in France in May -- Cremieu -- before flying out of Lyon, but your flight is just too early for that. I'm someone who never stays at an airport hotel, but your situation may warrant that. You can always spend the day and have dinner in Lyon, then go to your airport hotel at the end of the evening; the Rhone Express runs until midnight You can also take the above advice and stay in the Part-Dieu area and take the first Rhone Express bus or a taxi the next morning. I've stayed near Part-Dieu; it's not very attractive, but perfectly fine.

Thank you, both, for your input. Yes, very early but, unfortunately, not much choice when we have to make a connection in Paris. We'll see how it goes.