What does the line, "Frailty thy name is woman," in Act 1 of Hamlet mean and how might it foreshadow events to come?

Renaissance thinking was that women were "hot," that is to say they were ruled by their passions (emotions) that would run amok if not kept in constant check (under constant control). 

What the phrase means is that women, to Hamlet's mind, represent all that is frail (breakable, delicate, weak) in human nature; all weakness is bound up in, epitomized in women, and Gertrude is the epitome of this womanly weakness and frailty.  So Hamlet is referring to his mother's weaknesses: morally, spiritually, and physically.  Morally she is "frail" because she betrayed her husband by marrying Claudius and had the indecency to do so a mere one month after King Hamlet's death: "and yet, within a month—". 

HAMLET. Within a month,
Ere yet the salt of most unrighteous tears
Had left the flushing in her galled eyes,
She married.

Spiritually she is frail because (in Hamlet's mind, as are all these judgments) she has committed an unforgivable sin by marrying Claudius.  Physically she is "frail" because she is a woman and less strong and robust than a man, though this obviously is the least of Hamlet's concerns.

If you could do some research on this Renaissance idea of women, it would help you to explain Gertrude's "frailness" and round out your paper.  I've included a link below to get you started. 

The line about frailty might be said to foreshadow coming events because, with Hamlet's emphasis on frailty, he is proved to be himself frail by his own inability to understand the nature of the Ghost (Is it his father's true ghost or a demon sent from hell to ensnare him in evil?) and his inability to understand the role of revenge regicide (king killing; Claudius is now King) in a Protestant Prince's royal role and personal choices. These inabilities preventing coming to an understanding of complex events around him is what causes his famous inaction, hesitation, indecision.

It also might be said to foreshadow the frailty shown later by Ophelia when she is dumbfounded and driven to madness by the combination of complex events around her that she is unable to understand starting with Hamlet's strange behavior and ending with Hamlet's blind-sighted murder of her father, Polonius.