The Barnwell Center is five whole minutes of fun. It usually has an art exhibit in its small gallery and the gardens in the dome are pretty. But the building is small.
I would recommend taking a stroll around the area and during the stroll, take a small detour inside- it is entirely free after all. Other places nearby are the casinos, and even if you don't like gambling they have ice cream, and Sci-Port. Also, across the Texas Street Bridge is the Boardwalk, which is a nice area to walk around on sunny days and warm nights(also is safer than walking around downtown at sunset :) ).
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