In English, the word "mom" usually has a possessive pronoun before it.
you should greet your mom after she gets home from work.
So, you we usually write "my mom" or "her mom" or "your mom" instead of using the word "mom" in isolation.
In English, the word "get" sometimes means the same thing as "arrive". American-English speakers would use, "My mom had just gotten home"
Your original text said something like, "Mom had just been at home". That insinuates that the mother was at home in the past, but the mother is not at home anymore. The phrase "just been home" is not used to talk about people coming back home.
The child's mother had just gotten home, but her son didn't pay any attention to her. He just kept watching TV instead of saying, "Hello".
The proceeding block-quote is written in natural-sounding American English.
You asked whether or not it is it natural to say the following to your son,
“You should care about Mom by saying something to her when she's home.”
No, that is not natural-sounding English. The following is more natural-sounding:
“You should show that you care about your Mom by saying something to her when she gets home. For example, you could say ‘How was your day?’ or ‘Hello mom’”