What is the significance of this passage from Hamlet?
Gertrude is not only encouraging Hamlet to accept the death of his father, the former King, but perhaps more important, in the lines
And let thine eyes look like a friend of Denmark./Do not for ever with they vailed lids/Seek for the noble father in the dust. . . .
she is telling Hamlet to accept the fact that Claudius is the country's new king--Claudius, like Hamlet's father before him, is now "Denmark"-- and as such, deserves Hamlet's support. Implicit in her statement is the concept of feudal, as well as family, obligations to which Hamlet must adhere. And given Shakespeare's skill with and love of wordplay, Gertrude's reference to Hamlet's "vailed lids" may be a subtle acknowledgement that veiled lids indicate disguised motivations rather than constitute merely a physical expression of mourning.
The importance of Hamlet's situation is clear in his next speech to his mother when he answers her question, "why seems it [his grief] so particular with thee?" Hamlet points out that the trappings of grief are just that, but his grief goes much deeper than the outward manifestations suggest:
. . . These indeed seem,/For they are the actions that a man might play;/But I have that within which passeth show--/These but the trappings and the suits of woe.
Given Hamlet's suspicions about his father's death at Claudius's hands, it seems reasonable to believe that Hamlet is both describing the depth of his grief here and pointing out that perhaps the grief of others, most notably, Claudius, is characterized only by "trappings and the suits of woe."
Queen Gertrude instructs Hamlet to "cast thy nighted color off" in Act I, scene two. Gertrude and Claudius both approach Hamlet after Claudius' announcement that he has married Hamlet's mother.
Hamlet is visibly upset at this news, so Gertrude attempts to pacify him, telling him to "cast thy nighted color off and let thine eye look like a friend on Denmark" (I.ii.70-71). The meaning of these lines are two fold. First, Gertrude does not wish for Hamlet to pout angrily about the news; she wants him to be more friendly to the new king. "Cast thy nighted color off" also suggests black, the color of mourning, and that Hamlet most likely was dressed in black. Gertrude wants Hamlet to be able to look past the tragedy of his father's death and see the potential of the future.
"Seek for thy noble father in the dust.Thou know’st ’tis common. All that lives must die,Passing through nature to eternity" (I.ii.73-5).