Will Arya use the dagger to kill Littlefinger?
It certainly seems that way. Bran's comment -- "it doesn't matter now" -- before giving the weapon to Arya indicates that she's meant to have it, but might also hint that there's no need to worry about Littlefinger, as he'll soon be gone. As the Three-Eyed Raven, Bran isn't meant to meddle, a consequence he learned after the death of Hodor. But the resigned way he gives up the dagger implies that this will all happen anyway, and he's merely nudging along the process.
Arya has plenty of reason to kill Littlefinger. He betrayed her father, he sold Sansa to Ramsay. He's indirectly responsible for the fall of House Stark. Once Bran -- or one of Maester Luwin's documents-- reveals Littlefinger's role in the War of the Five Kings, Sansa will certainly order his death. Arya, with her thirst for vengeance, would be the perfect person to carry it out. And it would, in a way, absolve her of her kill list. By taking down Littlefinger, who in a sense is the reason she ever created it, our heroine will have avenged her father's death in one mighty swoop.
Even if she doesn't use it on Littlefinger, the catspaw dagger certainly has a major role to play. Valyrian steel has always been of value in Westeros, but its also one of the few things that will kill a White Walker. Now that winter is here, it's a pretty safe bet that we'll see some Arya-on-Walker action in the season to come.