I think if she just moved to the city regardless of if it's in Peter house or her own house it would be better option,then to just make her babysit you from the start it's just kind of weird ,
I think that's something that should be unlocked later on plus it be kind of cliche,it's better if the story for her revolves around something else then just babysitting to make it more interesting.
One thing I do suggest for anybody that decides taking up writing dialogue for mom or grandma please don't over do the affection they have towards Peter ,because then it just makes it seem like they treat Peter like a little kid,they should be just normal towards him no cute nicknames or mushy stuff like that or it just gets kinda cringy.