I'll be in Lyon for a few days in June (taking a train from Paris) and plan on renting a car to use around Lyon. It sounds like I can drive from Lyon to the French side of the Geneva airport and drop off the car. Is that possible? Should I take the train instead? My flight is at 12:30 p.m. on a Tuesday.
Actually the simplest answer to your question is Yes.
The airport of Geneva, although situated in Switzerland, is an international airport and has a French section. It can be accessed from France without passing the Swiss customs, through a special road (route douanière).
There are car rentals at the French section of the Airport, for exemple Sixt. This means you can rent a car in any Sixt Agency in Lyon (there is one at Part-Dieu train station), and drop the car at the French section of the airport of Genève, that you access via this road exclusively.
Here is an excerpt of the GVA agency of Sixt's web page:
As a side note, conversely, the Basel-Mulhouse airport is located in France, but is accessible from Switzerland via a similar road.
Warning: I don't know whether your car rental contract will allow you to drive on Swiss roads, so if that's not the case, beware that accessing the airport only via French roads can be a bit long, because Pays de Gex is almost an enclave due to the Jura mountains. You can count on your hands the roads that connect that area to the rest of the French territory, and therefore you can experience traffic jams when going there via Bellegarde.
If you want to go to Geneva by train it seems to be difficult: you'll have to board train TER 17601 from Lyon to Grenoble at 06:08 then TER 96608 from Grenoble to Geneva at 07:54 (21 minutes for transfer at Grenoble), you arrive at Genève at 10:00. 4 hours train for a short trip is really unexpected. There are Eurolines buses that depart from Lyon at 08:30 and arrive at Geneva at 11:30 on the timetable, but that's too late to catch your plane.
It's certainly doable. If you have a French rental car, you will need to either leave the motorway in Bellegarde-sur-Valserine and go to Ferney-Voltaire using the secondary road network or buy a CHF 40 motorway vignette for Switzerland as the car most likely will not have one. Train is also quite convenient but goes to the Swiss side, the duration of the journey from Lyon should be more-or-less similar. Within the airport, you can switch sides easily if you need to.
Do check which car hire companies are present in the French sector, some only have a desk on the Swiss side. You should also probably check in advance what they charge for dropping off the car in the French sector but in principle it should be a regular surcharge similar to what you would pay to drop it off somewhere else in France.
If you need to completely avoid going through Switzerland to reach the French side of the airport, it will be more difficult. You won't be able to take the train but could take a bus. With the car, you will need to pay attention to the directions; road signs and navigation devices would most likely bring you to the airport through Switzerland.
If you decide to rent a car, beware that not all rental companies will accept taking a car in Lyon and dropping it off in Geneva (a different country). Those that do will likely charge extra for it (in addition to the motorway surcharge).
Normally 12:30 should be plenty of time to catch a train from Lyon. The airport website mentions an 8:38 departure with a connection in Genève Cornavin, for a trip lasting a little over 2 hours, which sounds plausible. However, this page does not seem to be accurate: according to CFF, the Lyon–Geneva train is no longer running, and the only morning connection on a weekday departs from Lyon at 7:20 and involves taking a TGV (high-speed train — but not on this stretch, with compulsory reservation and market pricing) from Bourg-en-Bresse to Geneva. This is probably due to ongoing repair works on several lines in the northern French Alps and in the Geneva area. For the June period, the current announcement is:
The CFF page states that “More precise information will be available on this page shortly.” The SNCF page promises more information 2 weeks before the beginning of each phase, which for your trip will be on 21 April. The lack of accurate information probably caused some trains not to be listed in the database even though they will end up running with only a slightly modified schedule or with a partial bus substitution.
You may want to put off planning until more information is available. I expect that there will be a reasonable train or train+bus connection that will get you there in time. You may also want to research bus alternatives. The most plausible connection would be via Annecy (there is a 9:15 bus to downtown Geneva), but it doesn't seem that you can get there in time from Lyon. There is a bus that leaves Grenoble at 8:30 and Chambéry at 9:30 (in both cases, the bus stop is just outside the train station), arriving at Geneva airport at 10:45. SNCF currently lists a 7:32 departue from Lyon to Chambéry, but there is no schedule for June; there is a 6:49 departure to Grenoble. Even with the track works, there will probably be a more direct connection to Geneva that doesn't involve that bus. You might also consider driving to Annecy or Chambéry and dropping off the car there to avoid Swiss motorway and international drop-off fees.