Here are some ideas outside of restaurants and bars that are free or relatively cheap:
-Art Car Parade is going on this weekend I think, never been but looks cool from pictures I've seen
-Speaking of the parade, they usually have 5-6 cars on display at the Art Car Museum (free)
-Menil Collection and the park that's right across the street (free)
-Graffiti Park off 59 (can be sketchy but free)
-Walk around 19th Street in The Heights; White Linen Night is usually in August or early September. Great Christmas lights ("Lights in The Heights") in the neighborhood during that season
-Miller Outdoor Theater events (mostly free); I like going to watch the symphony when they play in the summer
-Houston Arboretum (free)
-Hermann Park and Centennial Gardens (free but parking might be difficult to find)
-MFAH Bayou Bend/Rienzi (not necessarily cheap but worth a look)
-Buffalo Bayou Park, Sabine Bridge for some great views of the skyline. There is a tour of an underground cistern as well but that costs like $10 or so
-The water fountain across from the Williams Tower
T's and G's