Agree with Monument Inn and Joe Lee's though I've only had their burger and loved it.
Seabrook Classic Cafe; we go for their fried shrimp tacos, grilled fish tacos, or fried fish po-boy. I also like their chicken soup. They are known for their breakfasts (weekends only) and their chicken fried steaks.
If your not bound to La Porte and Seabrook here are links in the area and includes some places in Seabrook.
Ate at Mainly Drinks and Main 101 in LaPorte and wasn't impressed with either. They were both okay but nothing more. At Mainly Drinks there were about 15 of us and no one said they loved the food, just a lot of "it's ok".
The Captain Bennie's is open and although I haven't been there, I always loved the one on the Gulf Freeway. It's in Deer Park actually. Corner of East and 13th, right off of 225 (East Blvd exit).
I agree with Monument Inn and Joe Lee's. At MI, try the pan fried shrimp or fried oysters. I love them both.
I would avoid the temptation of the Kemah Boardwalk, all chains owned by the Landry's clan. Try the Monument Inn off Texas 225, or Joe Lee's in Kemah.
For history buffs, or a great place to take kids, visit the nearby, (Monument Inn), USS Texas Dreadnought, the only survivng WWI and WWII Batlleship, or the San Jacinto Monument, on the site of the last battle for Texas Independence in 1836.