The dagger first popped up in an unexpected place: as an illustration in an old book in the Season 7 premiere, "Dragonstone," when Samwell Tarly is reading a book about the Long Night (i.e. the last time the White Walkers came), discovering that there is a mountain of obsidian under Dragonstone. While he's reading, we catch a glimpse of what appears to be the Catspaw Dagger on one of the pages. On that page we can see the following text:
The Valyrians were familiar with dragonglass long before they came to Westeros. They called it "zirfyl perzfyl" [sic] which translated to "frozen fire" in Valyrian and eastern tales tell of how their dragons would thaw the stone with dragonflame until it became molten and malleable. The Valyrians then used it to build their strange monuments and building without seams and joints of our modern castles.
When Aegon the conqueror forged his Seven Kingdoms, he and his descendants would often decorate their blades with dragonglass feeling a kinship with the stone. The royal fashion for dragonglass ornamentation soon spread throughout the Seven Kingdoms to those wealthy enough to afford it. Hilts and pommels were and are the most common decoration for dragonglass if too brittle to make a useful crossguard. Indeed, its very brittleness is what relegate it to the great houses and the most successful merchants.
This connects the dagger, which we already knew was Valyrian steel, to a deeper history of Westeros (beyond it being used to try and kill Bran). In "The Spoils of War," Bran wonders who owned the Catspaw Dagger, and the show pretends that the only way to take that question is "don't you remember this dagger," when the Three-Eyed Raven was probably asking a much bigger question. A Valyrian steel-forged dagger, as Sam's book points out, is for the Great Houses like the other Valyrian Steel in Westeros... so how did it end up in Littlefinger's hands before the series began? Baelish owns no castle and made up his own House all for himself, the dagger wasn't handed down to him by any legitimate means.
Now we know that Baelish's possession of the dagger was just a means to the end of having it wind up in Arya's hand. During the Battle of Winterfell, after wights destroy the cool double-spear weapon Gendry made for her, Arya still hangs on to the weapon, knowing (thanks to Melisandre) how she has to use it. Just when it looks like the Night King has stopped her charge at the Godswood, though, her assassin training pays off -- she drops and catches the Catspaw in her other hand and jabs it into the undead leader, shattering him and his entire army, saving Winterfell and the living in the process.
So, yeah: The Catspaw Dagger turned out to be super important, just as we suspected. Now we'll see if Arya gets to use it against Cersei and her troops as the battle for the Iron Throne returns to its full force.