usually don't do theories because I like to be right and there's a good chance I won't be, but this came to me this morning and I'm sticking it out here, long before Book 7, where everyone will have a chance to forget it if it's wrong, but I can still smugly link to it in the small chance it's right.
It's about that locket, the one that Harry and Dumbledore find in the Bird Bath of Death but turns out not to be the real Slytherin Horcrux!Locket but a fake left with a "nyah nyah" note by one RAB. Fandom seems to have decided that RAB is Regulus Black and the real locket is that one the gang found in OotP at Grimmauld Place and couldn't open.
That makes a lot of sense to me, and my theory assumes that that is true, which of course it might not be. My theory, building on that, is this. When Dumbledore first comes to see Harry in HBP he tells him Sirius left a will, and that Grimmauld Place is now his. Now, we assume that the horcruxes being tracked down and destroyed will be the plot of Book 7, so we're talking about a big scavenger hunt. Things should presumably stand in the way of all the objects, so it's a puzzle. The locket is the object we already know we have to find, and there are already hints in HBP about the things that might stand in our way, namely that Mundungus may have stolen and pawned that locket.
But here's another thought. When Dumbledore tells Harry about the will he also says that he's not sure if the will is going to hold up, because knowing the Blacks, there's a good chance they might have some enchantment on the house so that it will rightfully pass to the eldest living Black, Bellatrix, no matter who Sirius wants to leave it to. He calls Kreacher to test if Harry can give him orders. Harry can give him orders, which not only means the house is his, and that Sirius "knew what he was doing" (perhaps knowing all about those types of spells, as a Black) but that Dumbledore has just given us information that went nowhere--for now. There was no reason for Dumbledore to give us that information plot-wise, yet. That test with Kreacher wasn't necessary to get Kreacher to Hogwarts. Dumbledore could easily have just said oh, and Harry you've also inherited Kreacher (pop!), maybe you should send him to Hogwarts. Then Harry could do that. The scene would be just as funny, certainly. I think that whole "test" idea could be there to lay the groundwork for a future test of the same type that somebody fails.
Since Dumbledore *did* suggest that kind of enchantment, I think it may come into play on that locket. If Regulus is the one who stole the locket and took it back to Grimmauld Place, *he* may have placed that kind of enchantment on it. Perhaps that is why no one could open it in OotP--the diary seemed to have no such spell on it (though of course Voldemort could have gotten more paranoid with later objects).
If that locket has a Black spell on it, that leaves a few people who are Blacks to open it: Sirius, Tonks, Narcissa, Bellatrix and Draco. Sirius was considered a traitor by his family, and there's good reason to believe there was no love lost between him and Regulus, so Regulus may not have considered Sirius a Black and not given him access to the locket. That leaves Tonks, but Tonks is a half-blood, and Regulus seems to have ascribed to "Pureblood Mania." Perhaps he'd turned against Voldemort, but he might still automatically rule out half-bloods.
That leaves the locket possibly being opened by Bella, Cissy (I love calling her Cissy) or Draco, all of whom make more dramatic sense because there's a good chance they won't do what our guys want, unlike Tonks. Bella obviously wouldn't do it. That leaves either Narcissa opening it to save Draco or vice-versa. I think Draco is obviously the more dramatic choice. It gives him something to actually do to save his family and himself. It would do something to make up for the pain he caused in HBP, even if it was under duress. Also, it much more neatly ties up the hints in OotP about having to work with Slytherin (which Harry thought was impossible because that meant Malfoy), and it would give a Slytherin a chance to change the character of that house. Voldemort used Slytherin's locket for his own purposes, so this would be a good way of symbolically showing that Slytherin=/=Volemort.
After all, the one other Founder thing we know of is the cup of Hufflepuff, and there's a possibility this might mean working with Zacharias, if only because it was owned by a Smith who claimed to be a Hufflepuff relation, and HBP made a point of stepping up the Zacharias Hate. Not that I think both stories (locket and cup) would be too similar, but they might need some information from Zach about his family history or some such.
Anyway, so that's my main theory, that House of Black is important to the story in the form of Regulus and that type of enchantment that Dumbledore described in HBP. One thing that always seemed really clear to me in OotP was that we should *not* take Sirius' words at face value in that chapter. He'd dismissed his family outright and assumed anything they did that didn't seem wasn't completely evil was a fake. His story about his brother already seems like it could be totally off. But also, it was Sirius' family that killed him, and just as it makes sense to "heal the house of Hogwarts" by having the houses sort of come together, so it would also be nice to "heal the house of Black" by using one of its "grey" members to finish Regulus' job and thus do something for Sirius.
ETA: I totally forgot Andromeda! Though she, too, may be out of the question for being a blood traitor.