What is the best way to travel from Lyons France to Monterosso al Mare?

There doesn't seem to be a good option here. Faced with this situation, I'd either drop the Cinque Terre or develop a burning desire to see something along the train route and break the journey there.

You can explore a bunch of possibilities on Rome2Rio.com. Alas, I have found that you absolutely cannot trust the trip durations, frequencies or fares shown on that website. However, it's pretty good about whether a particular air or rail route exists. If you want to look at flight schedules, try skyscanner.com. For train schedules I find the Deutsche Bahn website easiest to use, though you'll need to go elsewhere to buy the tickets.

In case you are not aware, the estimated 6-hour drive comes at the price of a likely extremely high international drop fee for picking up a car in France and dropping it in Italy. And of course that time (even if it is accurate--check ViaMichelin.com for a second opinion) doesn't include time spent picking up the car or detouring to a drop-off point and then finding your way to Monterosso. So the time savings would probably not be very significant in the end.

takes forever

Most would go via Turin, but it's not fast either.
You can take the TGV french train to Torino Porta Susa station that departs at 8:31 (3h52m); then you have 2 choices.

You can use the local subway to go from Turin Porta Susa to Turin Porta Nuova station and then take a train to Monterosso with a transfer at Genoa Brignole station.

Lyons > Turin PS > Subway > Turin PN > Genoa > Monterosso

Or you can take an Italian high speed train from Turin Porta Susa to Milano Centrale (55 minutes) and then a direct Intercity train from Milan to Monterosso (3 hours). This way, if everything goes right, you leave from France at 8:31 and arrive at Monterosso at 19:05. At least you avoid the transfer between different railway stations in Turin.

Lyons > Turin PS > Milan Centrale > Monterosso

I'd just fly.