It is notable that it is Juliet who first proposes the idea of marriage. Though still very young, even she knows that a woman in her position needs to protect herself, needs the security and respectability that marriage can bring.

Deep in her heart, Juliet may believe that Romeo's intentions are honorable. But she still needs an outward sign that Romeo's serious. Raising the prospect of marriage is very useful in this regard. It can be seen as almost a test of Romeo's intentions towards her, a way of making sure that Romeo really means what he says.

If Romeo is as deeply enamored of Juliet as he says he is, then Juliet is certain that he will have no problem with making their relationship official by agreeing to marry her. If Romeo's intentions are genuine, if he doesn't regard Juliet as just another infatuation, then he will have no hesitation in sending word the following day that he is ready to marry her:

If that thy bent of love be honorable,

Thy purpose marriage, send me word tomorrow. (II, ii, 143–44)

If he should do so, then that will put their relationship on a much stronger footing. Though it will still be a deeply intense, passionate romance, it will also have the firm foundation of a commitment to engage in the sacrament of holy matrimony.

Juliet says that if Romeo swears his love to her, she will believe it, but she also notes that words are unreliable, stating,

Yet if thou swear'st

Thou mayst prove false.

This is a really interesting part of Act II in Romeo and Juliet. The reader sees prior that Romeo has a predisposition to fall in love quickly and furiously, as seen in his very dramatic behavior after his falling out with Rosaline. Up to this point, a case could be made for Romeo simply being in lust with Juliet instead of actual love; the two haven't known each other long at all - only a few moments actually. It's reasonable to question both of their motives, but especially Romeo's based on his prior behavior. However, then the idea of marriage comes up the very evening they meet by Juliet: 

Three words, dear Romeo, and good night indeed. If that thy bent of love be honorable, Thy purpose marriage, send me word tomorrow, By one that I’ll procure to come to thee, Where and what time thou wilt perform the rite, And all my fortunes at thy foot I’ll lay And follow thee my lord throughout the world.

This is somewhat surprising, because they've just met. It's also shocking because it's Juliet who suggests this. Even today, society dictates that the man propose marriage. In Shakespeare's time, it's even more surprising that Juliet is the one who proposes this idea. 

Juliet proposes the idea of marriage to Romeo on the very night that they meet.  In Act 2, scene 2 (otherwise known as the balcony scene), Juliet becomes so obsessed over the idea that she must be sure that Romeo is truly in love with her that she tells him that if he really loves her and wants to be with her, then he must marry her when she says,

     "Three words, dear Romeo, and good night indeed.
      If that thy bent of love be honourable,
      Thy purpose marriage, send me word to-morrow, 
      By one that I'll procure to come to thee,
      Where and what time thou wilt perform the rite;"  (A. 2, s. 2, lines 148-152)

On the following day, Romeo does, in fact , give the Nurse information about when the marriage will take place.

The first discussion of marriage involving Juliet is in Act 1 Scene 3. It is Lady Capulet who suggests this notion to her daughter, and asks her to consider Paris as an appropriate match. Juliet says she had not contemplated marriage. The discussion involves Juliet, her mother and her nurse, emphasizing how important the nurse is to life of the Capulet family; and Juliet in particular.

It is in Act II scene ii when, bewitched by the intensity of her feelings for Romeo, Juliet asks him if he is true in his intentions, and is committed to marrying her.

If that thy bent of love be honourable,
Thy purpose marriage, send me word to-morrow,

It is likely that the notion would not have occurred to her so readily if she had not been discussing marriage with her mother and the nurse earlier in the evening.

Juliet is looking for affirmation that Romeo’s purpose is the same as hers, and that his passionate words are based on real emotion. The suggestion that they marry is her way of testing his motives. Both become excited by their daring plan.