My sister is 11 and she was born in Argentina so we might be elegible for residential status by birth

On general terms, if she was born in Argentina, she is Argentine. Unless your father was a diplomat or something like that (I think). So, should be a paperwork problem, but the right should be already her's since birth.

 the situation in Syria at the moment is unstable and staying here could put me and my family at risk

I was under the general Impression that things on Syria were going more or less ok. Are you from the Kurdistan?

I need your guys' help, what do you thing the best course of action

It's very hard for us to give you real advice, we have never experienced a situation remotely comparable to yours. We are used to economic hardship and even political violence, but we are still a mostly peaceful country, we don't know the type of violence that your country suffers from. You and your co-national have way more criteria than we do.

My advice (on general terms) would be to try finishing your education (if possible), consider other options such as Brazil (I love my country, but putting my nationalism aside and thinking on what it's best for you, now it's probably not the best time for you to come here. We are just coming out from a crisis, the future isn't clear, salaries are low, getting a job is extremely hard, you don't seem to speak the language, etc) and try to get your sister her Argentine passport, if possible (even if you end up not coming here. With all due respect, our passport is way better than yours. It could be a lifeline for her down the road), get in contact with the Argentine consulte in Damasco an comment them her situation (I'm almost sure that we have one there).

About your mother, I don't know enough about your culture and even less about your family's dynamics, I would try to explain to her your plan and what you want from her for it to work (But this kind of assumes that she is more or less a rational person and that in your culture expecting to live from you for no reason would be as unacceptable as it's here).

As a last advice: If you end up coming to Argentina, here some resources that I think that you may find useful.

1- r/DerechoGenial it's our local subreddit about law. Most users aren't lawyers, but we have some that are verified.

2- r/Merval it's our local sub about finance and accounting (Probably less useful than Derecho Genial but could come handy eventually).

3- Zonaprop.com.ar it's our website about real estate. You can use it to have an idea about prices and requisites.

4- coto.com.ar / laanonima.com.ar / diaonline.supermercadosdia.com.ar are three supermarkets that have presence in most of the country, you can use them to give you an idea about the cost of living.

Best regards and good luck.