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:
Notre agence se trouve au niveau 1 dans le hall des arrivées. Le parking P20 se situe au rez de chaussée du bâtiment abrittant (sic) le secteur France.
Le retour de votre véhicule doit s'effectuer au parking P20 au Secteur France de l'aéroport de Genève à l’adresse ci-dessous: Route douanière 01216 Ferney Voltaire Coordonnés GPS: N 46°23.97 E 11°84.08
There are other car rental companies: Europcar, Avis, Hertz for example.
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.