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I am jealous! Houston is great for eating big, that is for sure.... some of our favorites:
BBQ - Goode Co BBQ on Kirby
TexMex - Chuy's or Pappasitos
Upscale - Mark's, Cafe Annie's, Tony's and my favorite Brennan's
Ice Cream - Amy's in Montrose area
Most restaurants will be closed on Christmas as Houston, in general, is pretty religious. I would try the Goode Co. family of restaurants (they also have seafood, etc.) to see if they have any holiday take-out packages. You might also try something like The Capital Grille.
As for must-dos, admittedly there are not many..... I would suggest ice skating at the The Galleria, The Menil Collection, The Houston Zoo (small, but beautiful and inexpensive). You could also drive to Galveston (~60 min south) but the beaches are not anywhere as nice as they are here.