You could consider Cap Farret, the Villa Ephrussi, and lunch on the point, but I would still do Nice. Locals enjoy its old town! The views from the top of Eze Village are spectacular, and I haven't found the crowds to be overwhelming in October. That's when we usually travel and I've enjoyed my time in Eze, even during the mid day when we advise people to avoid the village during the tourist season. You might consider taking the train to Eze sur Mer where you catch bus 83 to the village and then take the 82 bus to Nice. Start out early and you should be fine with these two stops.
Here are links to the bus schedules. Check the the 82 and 83 for your connection to Eze Village and the 81 and 100 (as LondonBob suggests) for your trip back to Villefranche. You can also take the train back from Nice...whichever works for you.
I make copies of the PDF's and carry them with me.
For information and your peace of mind, here is a link to the train schedules. Follow the directions below the link to get to the PDF.
Under the second option, "Search Online," select the "O4 Mandelieu-La Napoule...Ventimiglia" route. Click "Select Time Sheet." and an underlined link will appear below "Please Choose a Line." That will get you to the PDF.
This is just so you know that you can do this, and you can pick up a colored brochure of the "04 Madelelieu-LaNapoule...Ventimiglia" schedule at the train station. Trains run frequently and are "walk on," meaning there aren't seat assignments and reservations aren't necessary. The trains' marquis will read its final destination, but as long as your destination is before that stop and the train stops at your intended station, you'll be good to go. Pay attention to the day schedules at the top of the schedule as there are some route changes on the weekends...you're always safe with "tous les jours!"
Have a great trip!