I don't teach children but I am pretty sure that it would not be standard pedagogical practice to attempt to teach the grammar behind this rule to a 9-year-old. Personally, it seems like a really bad idea to try it.

First tell her, "One uses the gerund form of the verb so that it may function as the object of 'stop'." After she stares at you blankly, tell her, "That's just the way it is." ;-)

Just tell her that STOP is a verb which tells you(understood but not named) to quit an activity. Grumbling is an activity. Walking is an activity . Eating is an activity.

Well, I agree on what has been said above. However, you could try to emphasize the difference during some activity. EX; Let´s make up a story. She his walking along the beach holding a camera. You say: Stop TO take a picture. --- She needs to stop for a while and take a picture. We need to stop TO recharge the battery. You say: Stop taking pictures, the battery is low. --- She needs to STOP taking pictures and put away the camera. If you stop talking while taking pictures and concentrate, the images will be a lot better. Practicing little by little she will link the difference to its meaning.

Hello Smelval. Stop TO DO - it means you stop whatever you are doing in order to carry out other activity. EX: We got lost so we stopped to take a look at the map. I just stopped to have lunch at 3pm. I was driving home and needed to stop to answer the phone. STOP DOING something - it means you give up, finish, quit something else. EX: She stopped crying. Stop CRYING, please. Stop DOING that. He stopped ACTING like a child when she went away. He stopped smoking 2 weeks ago. I hope it helps. :)