You'll hear this slightly old-fashioned, however quite common, expression mostly in the context of an event or an invitation. C'est à la bonne franquette! is a way of saying that the event is simple and informal, with no negative connotation. If food is shared, it will be a country-style, no-fuss meal. It might even be a potluck, although there is no real French equivalent to the concept of potluck itself, where every guest brings a dish to share. The closest you'll get to a potluck in France is when guests volunteer to bring a starter or a dessert. If it's à la bonne franquette, then you won't have to get up at 5am to get started on your multi-step recipe; a simple quiche or tarte aux pommes will do just fine.
The word franquette is a diminutive of franc (frank, straightforward), and is found only in this expression.
We find traces of the expression à la franquette as far back as the 17th century in the dialects of Normandy and Northern France. At the time the expression meant frankly, in all honesty. It gradually developed into meaning simple, probably in contrast to the polar opposite expression à la française, which was used to describe an event prepared with great efforts and sophistication, luxuriously.
Interestingly, both expressions derive from the same Germanic word frank. The word referred to both the Franks as a tribe - which would later become les Français - and the adjective frank, which meant free in Proto German. The Franks were the tribe of the free people.
When exactly did la franquette become la bonne franquette? No one knows exactly. As in many everyday French expressions, the adjectives bon/bonne or beau/belle are used to add emphasis and carry no particular meaning.
C’était une visite à la bonne franquette.
It was an unceremonious visit.
On va manger à la bonne franquette.
We’ll eat a simple meal.