Symbolism, Imagery, Allegory
This bad boy makes two appearances: first when the brothers are hassling the Duchess about remarrying in Act 1, and later when Ferdinand makes his really scary (and really rude) visit to the Duchess's bedroom in Act 3.
A Dagger for the Duchess
The first time, Ferdinand is trying to bring home his point about remarrying by taking out the dagger and saying to the Duchess,
You are my sister.
This was my father's poniard: do you see?
I'd be loath to see't look rusty, 'cause 'twas his. (1.1.322-24)
Ferdinand isn't just being a jerk and threatening to physically hurt the Duchess—his presentation of the dagger is framed by his presentation of their family structure: "Hey, I'm the brother, you're the sister, this is the symbol of our father, and this how our family works."
Their father may be dead, but his dagger represents the power that a patriarch typically held over his family. And that's exactly the power that Ferdinand wants to wield over his sister. Problem is, because of she's an independent widow, and, you know, the freaking Duchess, Ferdinand can't actually control the her in the way that most brothers of this era could control their unmarried sisters.
And Then There's the Erotic Subtext (We Knew You Were Waiting For It)
By threatening the Duchess with their father's dagger, Ferdinand is trying to oppress a woman who has uniquely escaped the normal constraints of a patriarchal social system with a symbol of patriarchal power.
That's not all, though—the dagger also serves as a phallic symbol, and lots of people interpret this moment as Ferdinand expressing his incestuous desire for his sister (go look at Ferdinand's "Character" section for more on this). Bear with us.
These readers firm up their position with the second dagger scene, when Ferdinand comes with the poniard into the Duchess's bedroom in Act 3 as she's getting ready for bed. Having discovered that she's given birth, Ferdinand gives her the dagger, telling her to kill herself now that she's ruined herself by being unchaste. Again, he's trying to dominate the Duchess's sexuality by threatening her with the emblem of the dagger.
And, yeah, it doesn't really fly.
If you think of Ferdinand as being all, "look at me, oppressing you with my man power-ahem-dagger," what does it say to you that the Duchess doesn't die of stabbing? It would have been just as easy for Webster to trot out the poniard for a final performance in Act 4's execution scene, but instead he chooses to have the Duchess strangled by a noose that's actually associated with her wedding ring (look at the "Rings" section).