Depends on what you want out of your trip. There are hotels around Vandy which, I assume, will be pricey. You can check the Green Hills area. It would be an easy Uber or drive (5 minutes) to Vandy’s campus and you might catch a bit of a break.
The cheapest hotels in Nashville will likely be out by the airport. But, that’s on the complete opposite side of the city from Vandy.
I like Franklin as much as Nashville. If you want a nice, quiet night out check hotels down there. They will likely be lighter on your bottom line, especially if you stay out on 65. Plus, it’ll give you a head start headed back south. You might not run into as much of a markup for the Titans Game out there too.
If you’re wanting to do the Broadway / Downtown, it’ll be pricey.