Clear Lake - head to Kemah Boardwalk, they rebuilt the amusement park after Ike.
Houston Space Center
Museum of Natural Science/Houston Zoo/Children's Museum/Fine Art Museum
Menil Collection
Galveston - check out the beach, Victorian area around The Strand, Segway tours, tall ship, Moody Gardens (botanical/butterfly/IMAX) and SCHLITTERBAHN, historic Hotel Galvez, also the spa and central pool at the San Luis are pretty nice. Drive down to Jamaica Beach.
Fishing (rentals and charters out of Galveston or Kemah)
History/Hill Country day trip: driveup to Washington on the Brazos, there's a museum and historic farm, terrific bluebonnets, eat at a BBQ joint or Bevers Kitchen in cute downtown Chappell Hill or sandwiches and pie at Must Be Heaven in Brenham or swing through Roundtop to look at antiques.