In "Hamlet" how does Ophelia's weakness contribute to the play's plot?

Ophelia tends to follow the lead of others, obey their will for her, and not stand up for herself.  At the beginning of the play, we see both her father and her brother lecturing her about her relationship with Hamlet; they tell her to stop seeing him, to be careful, and warn her that there is no way Hamlet will want to marry her and that he is probably just going after her for a little action.  Put yourself in her place--you love your boyfriend, a lot, and have gotten all indication from him that he loves you too.  Then, your family corners you and lectures you on his bad intentions, tries to convince you that it isn't love, and to stop seeing him.  I would have been pretty upset, and defended Hamlet, defended his honor, and tried to explain the situation better.  Not Ophelia.  Her response?   "I shall obey, my lord" (I.iii.136).

This weakness of hers starts to impact the plot of the play when she indeed obeys, and

"as you did command, I did repel his letters and denied his access to me" (II.ii.180-110).

She indeed, just as her father told her to, rejected Hamlet.  Hamlet, already disgruntled with the faithfulness of women, is only further embittered by Ophelia's seemingly fickle response to him.  He loved her, and all of a sudden, she rejects him.  It gets even worse when she agrees to, of all things, confront Hamlet in front of the king, queen and her father, just so they can observe Hamlet's reaction and study the two of them like they are interesting animals in some experiment.  She gives Hamlet his letters back, which prompts Hamlet's infamous rant against women, probably aimed paritally at her and mostly at his mother, whom he suspects is watching. Ophelia's actions only add to Hamlet's fury against women and his mother, and his harsh treatment of her adds to her stress, that leads to her mental breakdown later on.

Ophelia, used to never voicing her own thoughts and only living by the will of other people, is used as a pawn in other people's plans.  Her lack of backbone drive her to force away someone who truly loved her (Hamlet), and then, to go mad when all the people who were directing her life were gone.  Laertes is off on a journey with Polonius dies, and Ophelia, left without her meddling father's directions, becomes unhinged and directionless.  She cracks, and goes crazy.  This leads to her death, and then, to Laertes and his oath of revenge.  So, although Ophelia was not one of the most prominent characters in the play, her weak nature led to ke events.  I hope that those thoughts helped; good luck!