We go the the island often. Here are some of my favorites:
Breakfast: Yummies (plan on a wait), Josephine's Kitchen, The Cafe on the Beach, Grapevine Cafe, Karma has good coffee
Lunch/Dinner: Viva is on the bay, owned by Yummies' folks, Mahi Nic - casual spot by the boats on the south end of the island, Dirty Al's for fried seafood, Gabriella's for dinner, F&B if you're looking for upscale dinner, Liam's for steak and seafood, also a bit upscale.
Blackbeard's is going to get a lot of mentions because its an institution, but I'm not a huge fan, unless you're looking for a burger. The rest of the menu is too industrial for me. Tom & Jerry's has a good burger, casual dining. Meatball Cafe is pretty good for lunch.
As mentioned, Port Isabel if you're looking for Tex-Mex. Lots of options. I haven't been to the White Sands in years but it was consistent, especially the whole fried flounder or whole fried snapper.
Have fun!
People think I'm an idiot or something, because all I do is cut lawns for a living.