Hi
I have stayed at the MCO Hyatt a couple of times and really like it (although it is a bit "different" to a resort hotel). It is right over the terminal - you get into a lift in the terminal building and the upper floors are all the hotel. Both times we have had atrium rooms - the window overlooks the inside of the terminal: a bit wierd (and dark with no lights on), but interesting and great to people watch. There are rooms with normal windows, but we have never had one...
Rooms are nicely furnished and clean - I like them (and am a fussy person...).
There are a couple of resturants and bars there. I really like their casual restaurant, but - if you wanted something quicker and cheaper - it would be very easy to get something from one of the terminal outlets...
There is a small, but nice, pool overlooking the runways - a bit noisy, but different. There is also a small gym.
Personally, I would pick up a car the next morning. Am not sure what the parking facilities are like and why pay for the extra night (unless, of course, you want to get out and explore - in which case I would probably choose a hotel elsewhere in Orlando)?
What time will your apartment be ready? 5am seems a little excessive, but - if that's what you want to do - I would imagine that the hire car agencies are open for business that early and would allow you to pick up early. If I was you, I would go pick up the car myself (the hotel reception will tell you where to get the shuttle bus to collect your hire car - since the hotel is in the terminal, this wont take long or be much hassle), then go back and collect your young family and luggage.
I think that the airport Hyatt is a good choice for a first night.
http://orlandoairport.hyatt.com/property/index.jhtmlBoo