There were several museums that were not open on the weekend which looked interesting. The Fire Museum of Texas and Beaumont Police Dept. Museum both are located downtown. We also spent a few hours at Crockett Street Entertainment District on Friday night. This is new area downtown that houses several restaurants (all are not opened yet) and bars. We sat outside on the second level and enjoyed a drink. It looks like this will be a great place as soon as it's finished.
Sunday morning we went to Riverfront park. This is a nice location, close to downtown, that has a walking path including a boardwalk and a look out tower along the Neches River. There are picnic tables and a playground area. It's free to park there.