Juliet has a lot to lose, and that is significant in considering the internal conflict she faces. She is from an influential family, and her father has arranged a suitable marriage for her. As a woman in this era, she is forced via societal constraints to rely on the men in her life to provide security for her. There is almost no possibility that she could hope to sustain herself, especially not with the luxuries she has grown so accustomed to as a Capulet.
After falling in love with Romeo, she realizes she has a tough choice to make. To follow her heart means to follow Romeo and almost certainly be forced to give up all that it means to be a Capulet. Later she will find that she has even underestimated her father's reaction as he commands her to marry Paris—otherwise she will be effectively dead to him. Although often romanticized, Juliet in this scene needs to know with certainty if Romeo is worth the risk.
Before she even knows that Romeo is listening, Juliet ponders this issue aloud. She wishes Romeo would simply give up his name because his only fault is being a Montague. Apart from having the last name of her family's sworn enemies, she believes that Romeo is her ideal companion:
What’s Montague? It is nor hand, nor foot,
Nor arm, nor face, nor any other part
Belonging to a man. O, be some other name!
What’s in a name? That which we call a rose
By any other word would smell as sweet. (act 2, scene 2, lines 42–46)
Romeo ends up convincing Juliet that his last name means nothing compared to her love, and they begin making plans for a future together.