I often travel to Houston for medical treatment and am very familiar with the area. My recommendation is SpringHill Suites Marriott.
The rooms are all suites, with a living room with sleeper sofa and seperate bedroom. They also have a microwave/fridge/coffee maker, which can be quite handy. They offer a free full breakfast, free parking, and free shuttle to/from medical center. Medical center rate our last trip was $99, with tax came to $116 per night. Its a nice place to take kids, my kids always enjoy staying there. It has a fairly nice outdoor pool.
Do be aware that the area around the Houston Medical Center can be very expensive, for meals, parking, and hotel tax (about 18%). Look for a hotel with free parking, shuttle, and some opportunity to cook in your room. Most major medical facilities charge $15-$18 per day for parking!
Do also be aware that the area around the Medical Center can be a little grungy. Actually, the Medical Center itself is quite nice and safe, but there are some seedy areas around it. I would not recommend using Priceline or Hotwire for a hotel in this area, you really need to know your hotel.
You must call the hotel direct for a medical center/patient rate (the $99 rate I got). You can't access it via the website. They will want the doctor's name, phone number, and patient ID. They want that for a medical center rate anywhere in the area, just to cut down on abuse of the rate.
If you choose to stay at this hotel, but sure to askat the front desk for coupons for the pizza delivery service that delivers to the hotel. Some of the best pizza we've ever eaten, and quite a reasonable price! Also, check out the Houston Children's Museum, right in the area. They have a free night once a week (with free parking), however the free night varies, best to check with them.
Any questions about the area feel free to ask. I hope all goes well for you and your family!