If you can score a reservation, I would highly recommend a lunch or dinner at Peter Luger's steakhouse in Williamsburg. It's an institution and the best steak I've ever eaten. Also has strong old-school Brooklyn vibes.
At the very northern tip of Greenpoint, there is a lovely Japanese cafe by the name of Ashbox. Perfect for a light breakfast or lunch.
Also in Greenpoint is Glasserie, which is where the Madcapara duo (at the Grand Central Market) got their start and built their rep, order the rabbit three ways.
If you're willing to make the short hop north to Long Island City (which despite the name is not in Long Island but just north of Greenpoint in Queens - easily accessible across the Pulaski bridge), you can eat some good mexican food at Casa Enrique. If the weather is decent ask to sit outside in their courtyard.