Families most need very peaceful, private and quiet neighbourhoods to stay. The suburbs are the quietest place so far. The west side of Stuttgart is very quiet, it is near the city centre and the cost of living is not that high.
A good example is Degerloch neighbourhood which has a very family friendly environment. Stuttgart West is not to be overlooked too because it has some major parks, shops and enough entertainment places for the families.
As for hotels, you can get accommodation for your family in a number of places with utmost comfort.