I’m planning to work at home(sideline)through internet businesses like writing papers online and make a money. However, I’m handling a Dependent visa through my mom who has an H1 visa (working visa) in the US. They said that you can’t work with your dependent visa only but it’s based online. So is it possible?

If you are working for a US employer, the answer is a flat-out no. However, if you are working for, say, a Canadian consulting company or something, and doing some trade work in USA, you do not need a social security # (only a Canadian SIN #). You need to have a SSN/SIN for whatever country your employer is based out of and wherever you pay taxes.
I assume you are on a H4, you should have a ITIN so if you earn money on line, then you could file it that way. As on this visa you are not entitled to work or to a SSN. If you are a minor your mum will file you on her taxes with your ITIN number
Baby sit for cash only.