I have stayed in an Italian town for a month and it was a great experience but here are my cautions and thoughts:
"Sleepy" towns are either untouristed or off-season for tourists. As others have mentioned this means reduced transportation options and opportunities. Off season can also mean closed restaurants and attractions as Italians take their vacations like any place with a strong season. This also means how much Italian you speak becomes important because while English is pretty common in touristy places and hospitality it fades pretty quickly in the country and with older Italians.
The limitations on your Italian will also limit how well you can get to know the people. Italian love to talk but you can exhaust basic language skills pretty quickly. And not being able to follow the tales of their children and cousins limits how much you can talk about the the things they want to share with you. And especially if you are traveling alone this can isolating. As much as everyone hates to think of themselves as even temporary ex-pats an ex-pat community can really help you discover a place.
Italian towns also often have festivals and events and these can be great to be present for, but trying to book a place for a month can often overlap with a really booked up week or weekend that can ruin your plans. Or make them much more expensive than it would otherwise be. In late October/early November Lucca hosts a comic-con that brings 8x the normal population to the old town and the availability and the prices for that weekend are crazy.
Sleepy can also mean fewer places to choose from to stay but hopefully also lower prices.
My advice is to find the cuisine/culture than most interests you in Italy and start there. Regions heavily dictate cuisine in Italy and you're not a big fish eater booking a place by the coast will probably disappoint you with the choices. After that balance size with "enough" connections to the outside world and restaurant and attraction options with the small enough to "get to know it" town. You need enough restaurants to sample from while you're there to really discover new things.
Once you shed your tourist mode and downshift it is shockingly to fall into the gathering water, bread, a few vegetables daily routine to cook and just explore. I did one of the eight potential side trips I had possibly planned and the time passed amazingly quickly. I completely recommend it but it is not without its pitfalls.
Best of luck finding what you're looking for,
=Tod