You would more so say “because my mother is sick, I cooked dinner for her” or “my mother is sick, so I cooked dinner for her” “so” is what we call a coordinating conjunction meaning it is needed to connect the two parts. “My mother is sick” is a sentence that could stand on its own but because you want to add more to that “so” is used to connect them. I hope that wasn’t too complicated!

If you say 'My mother being sick, SO I cooked dinner for her.' with 'so', it is grammatically wrong. The reason is because of 'being'. Using the present participle (-ing form) this way creates a 'participle phrase' which can describe a parallell situation, a reason, and other things. If you add 'so', it is like saying 'Because my mother ... so I ...' using the cause-effect conjunctive twice (because, so). Another example: Being so rich, he can afford a ten-room mansion = As he is so rich, he can afford a ten-room mansion.

Yeah if you want the two sentences to be related to each other (i.e. the reason you cooked dinner for your mother was because she was unwell) you should add so in between Or start the sentence with “as or because....” My mother was sick so I cooked dinner for her. As my mother was sick, I cooked dinner for her.