To save a dime, we booked my Bro on a 6:55am flight back home. I don't own a car in Lyon, since I live in the center and a taxi could easily cost us 50€ one way. I routed and defined a way to get to the airport using public transports, so here's how it goes. It is still slightly pricy and can take about 45 minutes to get to the airport; but it's worth it if you're carless like myself.
First: What time is your flight?
Make sure you know what time the flight is, once you get that information work about 1.5 hours before that, meaning, plan to arrive by an hour and half prior to take off. Lyon St. Exupery is a really easy airport, the security lines are minimal and it's never taken me more than 20 minutes to get through baggage drop and security to the gate.
Second: Go to TCL, plan the trip...
I used the TCL's website trip planner to get me to 'Vaulx-en-Velin-la-soie'. This is an above ground and below ground metro stop; they have tramways in this stop. You can also plan to go to Lyon Part-Dieu train station. With any of these, budget in about 20 minutes, as it usually takes me that long to get from city center to Vaulx-en-Velin.
Lyon recently opened up a new tramway called "RhoneExpress" that will take you directly from Part-Dieu or Vaulx-en-Valin to the airport. Unfortunately the cost is a bit high, 19€ for a round trip, 11€ for a one way (12-25 years old). The tram runs at 5:10, 5:40 am and after 6:00am every 15 minutes. It's a fast way to get to the airport, and will take around 30 minutes from Part-Dieu and 20 minutes from Vaulx-en-Valin.
That's how I ended up getting up at 3:30 am this morning to drop my baby brother at the airport. He's not quite old enough to figure out the metro stuff himself so I had to accompany him to the airport.. a chunk of change later, lack of sleep and feeling like a zombie; he's on his way back to the states.
Good luck on transport to the airport!