I am 13: old enough to have social media, but my mom's phone got hacked, and she won't let me get an Instagram, how can I convince her that I am responsible and will be safe?

I personally believe 13 is too young to be on social media... So just listen to your mom then :p

This is so right that you're getting older, and you deserve space on your own and you need privacy and starting to choose your self paths but  you're not really old enough to have social media, you're barely in age of having a phone. (you barely in age of this community as well)

You wanna convince her ?

1.At the age of 13, what purpose can you possibly have for having social media account ? Provide your purpose and reasoning to your mother, see if they are truly convincing.

2.However that mothers will always be sensitive on their child, yet, your responsible attitude will be judged during lifetime! If you can't be safe and responsible in your daily life then be sure you can't handle a social media account. Prove you're a responsible child, citizen and member of the family, contribute and participate for the family, earn her trust and you'll have your prize.

Also, try to listen to your mother, she wants your good, we living in technology age and social media is a very useful asset but there's cons and trolls there just like anywhere else, one wrong step, you'll have something to regret for the rest of your life. So always include your mother and make sure to go forward step by step.