Head to Lumen Café if you'd like to avoid the tourist-heavy streets of the Jewish Quarter but still get a cup of specialty coffee or craft beer in a hip neighborhood. With egg-based breakfast dishes (served until noon), a full-service kitchen, and a sleek interior featuring plenty of greenery, concrete, and wood finishes, Lumen is more than your average neighborhood café. But it's the patrons, students, artists, and neighborhood bohemians, who give soul to the place.
One of the best kept secrets in Budapest is Lumen's concert hall—adjacent to the main area—where they host daily live music concerts featuring some of the leading lights of Hungarian jazz, folk, and indie music. Also, keep an eye out for the outdoor garden in the back of the space where you might be witness to deeply philosophical conversations fueled by alcohol.
Note that there are two Lumens—there's a smaller location around the block from here and operating under the same name—but this one, in Horánszky Street, is where most of the action is.