Downtown isn't bad safety-wise if you use your head. The concierge should be able to direct you to plenty of places within walking distance of the Hyatt. The light rail is pretty easy to use, too, but it is a limited route. There are lots of bars and restaurants along Main. Might require a cab, but between Dallas and W. Gray on the other side of I-45 there are lots of options.
I'm not downtown very often, but I like the Flying Saucer (downtown) for a good cocktail and the Ginger Man (Rice Village-probably have to get a cab or use the buses to get there, but if you check out the routes/schedules the buses are managable, but are what you would expect) for just about any beer available on tap. Vic & Anthony's, Post Oak Grill and the Strip House are decent places to get a steak, and I think there is a Pappa's BBQ down there, but I'm not as big a fan as others. I'm sure there are some good mexican places down there, but I'm not familiar with any. Kim Son is also supposed to be good. I like the habanero shrimp burrito at Cabo for like $7.
Kind of a stretch, but maybe check out what is going on at Discovery Green. Never been there, but occaisionally there might be a concert or something there that some people might find interesting. Astros are another option if they are in town, and the stadium is not far form the rail.
Check out the Aquarium or the food court at Ikea if you have time.
