Robin Clayton with Jackson in Brooklyn, where she recently bought a one-bedroom apartment. Her budget hovered around $1 million. Katherine Marks for The New York Times

Several years ago, Robin Clayton was renting a studio in a TriBeCa condominium when a friend in Brooklyn invited her to housesit in Fort Greene for a few weeks. Before long, Ms. Clayton had fallen for the neighborhood, with its tree-lined brownstone streets and neighborly vibe.