"Penelope" is one big allusion to Homer's Odyssey, the ancient Greek tale of the clever Odysseus's wanderings after the Trojan War. Penelope, Odysseus's patient and loyal wife, is a relatively minor character in that story. By foregrounding Penelope here, Duffy suggests that women often get short shrift in classic art—and that it therefore behooves them to make art of their own.
In the original tale, Penelope cleverly uses her skill as a weaver to fend off a crowd of suitors during Odysseus's long absence. Faithful to her husband and unwilling to give up on his return, she tells the suitors that she'll marry one of them as soon as she's done weaving a burial shroud for her father-in-law—then unweaves each day's work every evening when they're not looking.
In Duffy's version of the tale, Penelope sews and unpicks her embroideries, not so that she can wait for Odysseus, but so she can buy herself more time to sew! Her dutiful "widow's face" is just a cover for her desire to dedicate herself to her own artistry.
That idea gets even clearer when Odysseus returns home at last. But this "far-too-late" return doesn't even make Penelope look up from her work. Instead, she "lick[s her] scarlet thread" and pokes it right through "the middle of the needle's eye." This image of a pierced eye alludes to one of Odysseus's own feats: the gory blinding of the terrible Cyclops, a carnivorous one-eyed monster. Through her creativity, then, Penelope has become her own capable, clever hero; she doesn't need Odysseus around anymore.
The poem's allusions to the Odyssey (and its transformation of that old story) suggest that women's creativity can give them subversive power and freedom. Not only does Penelope get to tell her own version of her story, she gets to embroider a whole new life for herself.