Is it permissible for a mother to be angry with her son because he plays video games for entertainment, and what is the ruling on her anger and her refusal to reconcile with her son?

If the games are permissible and do not include anything forbidden, then the 기본 rule is their permissibility. However, you must obey your mother if she prevents you from playing them, because she desires what is best for you. Therefore, seek your mother's permission before playing and try to please her by reducing the duration of your play and by showing filial piety, so that your playing against her wishes does not become an act of forbidden disobedience.