Sure . . . and vice versa! It just depends on what doors you’re interested in. Do you want to spend the rest of your life in the San Francisco Bay Area? Then Stanford’s the place to be! Plan on returning to Texas to be close to your family? Then Rice is going to offer you far more opportunities than Stanford. As for the rest of the country - anyone who knows anything (both grad schools and employers) is familiar with both Rice and Stanford and is going to have enormous respect for a graduate from either institution. Once you leave the country, things change - Stanford has far better name recognition worldwide . . . but does that matter to you?
Sure, visit Stanford and see what you think . . . and if, after visiting, you want to come home to Rice, then that’s exactly what you should do! 
But, if you decide on Rice, do talk to Rice’s financial aid department about the possibility of increasing your award. You can show them the Stanford award letter and ask them if they can meet it, even half way, just to make Rice more affordable for your family. And tell them Rice is your first choice.
P.S. Don’t count on being able to transfer back to Rice if you decide on Stanford and later change your mind. You can always apply, but there are no guarantees.