|Sleep little dumpling. I have replaced your mother.|
Let us broaden the perspective of having a robot in the family home to not just a nanny - who has sole charge and is entirely responsible for raising the children, and should be human - but a carer for members of the family of all ages, fulfilling multiple roles. The robot would need to be pleasant to interact with, human-shaped, upgradeable/programmable and reliable. But not a 'person'. If they were so self-aware that they were effectively a person, it would become immoral to make such demands on their time and refuse them the opportunity for self-development. Although the development of such robots as citizens can be an interesting societal development, such as in Iain M. Banks' Culture novels, it would probably be cruel to develop them as slaves.
Children and babies need human interaction in order to form healthy attachments and learn human behaviour. I was surprised to learn today that babies do not even pick up language sounds from tv or audio exposure but only from live humans. However, children don't need constant human interaction. Even while under supervision from good parents, a lot of a child's day will be spent not interacting. While pushing the pram, driving the car, cooking a meal, checking email etc. some or all of a parent's attention will be on that task and not on the child. It's possible and a good idea to chat to the baby while pushing them along or letting children help with the cooking, but it isn't possible to do it literally all the time. As long as the main carer for a child is a human who has freely chosen this role, a robot could take over some more menial or routine tasks, or pitch in for emergency care.
Especially in a context of disability or illness, a robot could be an invaluable helper. For example a parent who has difficulty walking or moving around outside may still want their children to be able to walk or cycle to and from school. Or a parent may need help with carrying and lifting a small child. A robot could also provide a disabled child with physical support (pushing a wheelchair, turning spoken sound into written subtitles...) and with support for behaviour and learning difficulties (informing others about the child's needs, reminding the child to stay on task, practicing flash cards with infinite patience...). All of this would allow both disabled parents and children greater autonomy. Some disabilities and illnesses come with good and bad days. When hiring a human assistant, you can't cancel or call them at short notice depending on how you feel that day, but a robot can do as much or as little as needed.
So what would a good robot need to be like? They need to be pleasant to interact with. If they sound like the self-checkout machines at the supermarket - "There is an unexpected item in the bagging area", they'd be so irritating they would never catch on. And they can't be like an automated telephone call either - "Say 'yes' to continue" or even worse "press '3' for more options". Interacting with the robot must be as natural as possible. It must know it is being addressed, and it must be able to 'translate' any number of different ways of phrasing an instruction. A certain level of command code might be acceptable, like the computer in Star Trek.
A family assistant robot needs to be human-shaped and -sized. Not only because that would make them socially easier to interact with but also for practical reasons. They have to be able fit through doors, walk up and down stairs, press buttons on appliances, plug cables into sockets, hold a mop, sit in the driver's seat of the car... They should also combine this human shape with superhuman strength and lightning reflexes.
Finally, a good family robot should be upgradable or programmable. As your family grows or your needs change, the robot should be able to learn new things, like for example sign language, or driving. Installed features on mine would include: driving licence, gps tracking, remote avatar capability (children and parents not always in the same country - this way dad could attend parent's evening etc.), phonics songs and backyard games, cooking and cleaning, ...
All the above of course assumes a level of technology we do not currently enjoy. Like the flying car, there have been advances and even prototypes, but we're just not that close to being able to build a machine that smart and that reliable. Parents are still getting in trouble with police for leaving children in charge of teenagers, so there is no way a robot at current levels of technology would be acceptable at this point. So, for now, this sci-fi geek will carry on calling on friends and family for those times when support is needed.