Saturday is market day in Beaune - skip the restaurants and put together some cheese, sausage, pate, etc. for a picnic you’ll never forget.
We had a late lunch on our arrival in Beaune at La Buissonniere and enjoyed it. A lot of other places were closed since it was around 2 PM. Small, casual, very local fare. Friendly, not too expensive.
We had a great lunch at La Table Du Square in Beaune when we were there this past spring. Very good food for not a lot of money, solid wine list including some by the glass options that had a bit of age.
le comptoir des tontons, as mentione by others, is also a great place to dine.