To support the argument, "Fraility thy name is not woman," a viable case can be made with the personage of Gertrude, the queen and mother of Hamlet. For, she does exhibit a keen intelligence, integrity, and fortitude in various scenes of the play. enotes states that critics who support her contend that she offers pithy and terse remarks in the play that "reflect her ability to grasp the magnitude of various situations."
In Act I, as a concerned mother, Gertrude perceptively urges Hamlet to act as the Prince that he is,
Good Hamlet, cast thy nighted color off,
And let thine eye look like a friend on Denmark.
Do not forever with thy vailed lids
Seek for thy noble father in the dust.
Thou know'st 'tis common. All that lives must die,
Passing through nature to eternity. (1.2.70-75)
In Act II where Polonius enters to announce the arrival of Voltemand and Cornelius from Norway and to report that he thinks he has found the "very cause of Hamlet's lunacy" (2.2.49), Gertrude interrupts the long-winded courtier by urging him to provide "more matter with less art" (2.2.95). Then, in Act III as the players perform, as the player queen makes a long and passionate declaration of love for the player king, Gertrude remarks, "The lady doth protest too much, methinks" (3.2.230).
Gertrude's integrity is demonstrated in her admission of guilt and remorse at the end of Hamlet's tirade in Act III:
O Hamlet, speak no more!
Thou turn'st mine eyes into my very soul,
And there I see such black and grained spots
As will not leave their tinct. (3.4.95-98)....
Be thou assured, if words be made of breath
And breath of life, I have no life to breathe
What thou hast said to me.(3.4.213-215)
Certainly, in the final act, Gertrude displays fortitude as she heroically drinks the poison intended for her son in an effort to save him. When Claudius tells her not to drink from the goblet, she replies, "I will, my lord; I pray you pardon me" (5.3.268). That hers is act of strength generated from love for her son is evidenced in her subsequent remark to Hamlet, "Come let me wipe thy face" (5.2.271).
While Gertrude, Queen of Denmark, is weaker than Claudius and subject to domination, she does, however, demonstrate a perceptive intelligence throughout the play, and a strong will in the end.
Concerning Shakespeare's Hamlet, arguing for or against the prompt--Frailty, thy name is not woman--is a bit beside the point. I assume you're asking for evidence that women are not frail in the play, in an attempt to prove Hamlet wrong. One can argue that easy enough: Gertrude is trapped in a man's world and must marry to keep her position in society, and even then her position is not paramount. Claudius clearly rules. This doesn't make her weak, however, it makes a woman's role in a patriarchy weak. One can argue she is pragmatic and practical, not frail.
She repeatedly stands up for her son against Claudius, and walks a line, so to speak, between her husband and her son. Gertrude is gullible, of course, and is duped by Claudius in numerous ways, but that doesn't mean she's frail. And Hamlet even contradicts his own statement later himself, when he says that surely Gertrude didn't marry Claudius for sex, because she is past the age when that is important to women.
But, again, that's a bit beside the point. One can legitimately discuss this issue on two fronts.
First, when Hamlet says that frailty thy name is woman, he is revealing his state of mind. What's at issue is Hamlet's melancholy or depression, and his obsession with his mother's hasty and incestuous remarriage. Hamlet is projecting his personal view of what his mother has done onto his view of all women (this type of projection is a sign of depression). Because his mother is frail, in his view, all women are frail. This is faulty logic, of course, and demonstrates Hamlet's frame of mind. His obsession with his mother's actions are, perhaps, a bit unusual or unnatural. Hamlet has suffered the death of his father and the loss of the throne for himself. Yet, he appears to be bothered much more by his mother's remarriage. His statement that women are frail characterizes Hamlet.
Secondly, one can discuss Hamlet's statement from the veiwpoint of feminism. The issue is: does the play present women in a negative light? Are the women genuine human beings in their own right, or are they only valuable in relation to the men? Answers to these questions and others like them reveal what Elizabethan society's views toward women were--at least in theory and in part. This is a legitimate discussion point concerning Hamlet's statement.
Looking for evidence from female characters in the play that proves Hamlet wrong, though, is not really legitimate. Although the play is not in any way a realistic play (it's written in iambic pentameter, for instance), the characters are complex personalities. Though the play is not realistic, the characters' personalities are lifelike. As such, of course they are frail. Human beings are frail. But they can also be intelligent and creative and greedy and stupid, etc.
In other words, Hamlet's statement doesn't have to be disproved. That's not the issue. Hamlet's state of mind is the issue. But if the characters, including Hamlet, and the play itself, present stereotypical, prejudicial views and depictions, then that can tell readers about Elizabethan's beliefs about women.
For instance, we know today of course that female sex drives do not dissipate as women age, and about that, at least, Hamlet is dead wrong. But you won't find evidence of that in the play.
One way to argue this point is to show that "Frailty thy name is man as well as woman." Gertrude and Ophelia ARE weak characters. They find themselves in a patriarchal society, somewhat powerless. Gertrude must marry Claudius to maintain her position, while Ophelia is dependent on her father, brother, and to some extent Hamlet for her position When her relationship with Hamlet is broken, Ophelia's hope for a good marriage, and thus an optimistic future is diminished.
But the men in the play are also weak. Claudius is not strong enough to resist his temptation to murder his brother to gain the throne and his queen. When he confronts the wickedness of his deeds in Act 3, scene 3, he is to weak to give up "My crown, mine own ambition, and my queen."
Polonius is also weak. He will not stand by his own advice "To thine own self be true," but instead sells out his daughter to gain the king's favor. In Act 2, Hamlet rightly calls him a "fishmonger." Rosencrantz and Guildenstern are "sponges" who will spy on their childhood friends because they wish to gain Claudius' favor. You can continue down the line. Almost all the characters, with perhaps the exception of Horatio and maybe Fortinbras (which is debatable) weakly submit to temptation.
Even Hamlet is portrayed as weak and vulnerable. When the play opens, one of his first lines is "Oh, that this too too sullied flesh shall melt." He says that he is "pigeon-liver'ed and lack[s] gall." Hamlet also feels powerless. He is faced with a villain but he is unsure as to how to proceed against him. He feels "like a whore who must unpack my heart with words" (Act 2, Scene 2).
Maybe this is one way to attack the prompt.