Not sure about restaurants, but I regularly go solo to some small bars and breweries in PVD and have made a bunch of friends at those places. Food is usually more pub food or food trucks though, but I imagine the environment is more conducive to talking to people than a true restaurant is.

Saw somebody mentioned the beer club meeting at Moniker, and across the street is Beer on Earth and Slow Rhode, both nice small places with cool people. Also nearby is Kimi's.

Edit: to add What Cheer Tavern in terms of friendly divey-bar. That's my go-to spot and I'm not sure I've ever seen someone come in and NOT make friends (if you're outgoing enough to talk to strangers).