Moody Gardens (aquarium rain forest, 3-D IMAX, etc.)
Pleasure Pier
Stroll around The Strand area
Free ferry ride to Port Bolivar (park and walk onto ferry to avoid the wait)
Expensive restaurants:
Rudy & Paco's (Highly recommend)
Shern's (at Moody Gardens)
Inexpensive non-chain restaurants:
Shrimp 'n Stuff (fried seafood, poboys, etc.)
The Spot (burgers, etc.)
Original Mexican
Sonny's (place looks questionable but is a favorite local restaurant/bar and family friendly)
DiBella's (Italian)
Mosquito Cafe (bistro-type food - highly recommended)
Check out LA King's, a candy & ice cream place on The Strand.