The Brooklyn Museum has a suggested admission, I believe, and their collection is great. Coney Island is a ton of fun if the weather's nice -- great people watching, street food, etc, and there's a museum as well. Get a copy of Time Out New York when you're in town and see what free event they have listed. smorgasburg.com is a great afternoon if the timing works. Park Slope has some remarkable brownstones to gawk at if you're into architecture. I hear good things about the Brooklyn Historical Society, but haven't seen any of their exhibits. I'm a systems dork, so I find the Transit Museum to be a ton of fun, but if you're not into that...Ellis Island is always worth a trip, although the ferry is $12.
What subway lines will you be on?