Given the lack of visible teeth and that all the sounds she made were ones that are easy without teeth, and the one she doesn't, da, uses teeth and tounge, she's probably missing one or both of those.
so basically
mame = bean/beans (cuz japanese plurals are weird)
edamame are like immature soy beans
theres a kid saying mame, and according to my dictionary (i cant think of a case off the top of my head that is so) it can mean something like child as well (as a prefix)
theres a lot of punnery going on here basically