Should she stop her prayer because her family does not know that she is Muslim?

Praise be to Allah.

Firstly: 

We congratulate you for the blessing of Islam, and we ask Allaah to make you steadfast and to help you and strengthen your faith. And we ask Him to guide all your family and your siblings, so that you will be a means of their success. 

Secondly: 

If you are afraid to declare your Islam and fear that some harm may be befall you, or if you want to delay telling people until you have learned more about Islam and are able to call your family to Islam, there is nothing wrong with you doing that. There is also nothing wrong with your stopping the prayer if you are afraid that someone will see you. 

The basic principle is that when the Muslim has started an obligatory prayer, he is obliged to complete it, and it is haraam for him to stop it, unless he has an excuse. 

The fuqaha’ have listed some excuses which make it permissible to stop the prayer. These include: 

-Fear of being harmed by an enemy or a wild animal.

-Fear for one's wealth or property, such as if a person is praying and someone comes who wants to take his property, or he fears that his mount will run away.

-Stopping the prayer in order to save someone from a fire or from drowning, or fear that a blind man may fall into a well.

See Radd al-Muhtaar, 1/654; al-Mabsoot, 3/2; Kashshaaf al-Qinaa’, 1/380. 

If you fear harm to yourself if any of your family see you praying and find out that you are a Muslim, there is no sin on you if you stop praying at that time, then pray after the danger has passed, which is what you did.

 Thirdly: 

You should strive to learn the rulings of Islam and keep company with good believing women, who will help you to obey Allaah. Think about the best way to call your family to Islam and strive to do that, for they are the most entitled of all people to your good treatment and da’wah. 

May Allaah help us and you to do that which He loves and which pleases Him. 

And Allaah knows best.