The power of Beric and Catelyn comes from the Old Gods, not the Lord of Light, and Jaime will be next.

[Greywater-Watch]

Theory No. 1:

The different religions exclude each other, a priest of R'hllor requires fire for his magic; a COTF/Greenseer requires a Weirwood. In extreme conditions the presence of the one (Fire/Weirwood) inhibits the magic of the other.

• Thoros cannot have visions by staring into the fire when upon High Heart, as the Ghost of High Heart explains to Thoros.
• Melisandre needs to get past the spells around Storm's End (an ancient fortress build with the help of the spells of the COTF), before she can do her Magic of birthing a shadow.

Theory No. 2:

There is only one god. Those who follow either R'hllor or the old Gods are wrong to believe in their exclusivity. Fire, Weirwoods - they all allow strong Magic.

• Thoros can re-surrect Beric Dondarrion after his fight and death against Sandor Clegane in a cave under a Weirwood tree and in presence of a fire. He could have extinguished the fire, and the re-surrection would have worked as well, due to the presence of the Weirwood.
• Melisandre's magical powers are increased at the Wall due to the Magic within the wall. This Magic is most probably a Magic of the old gods (the COTF helped build the Wall).
• The Kindly Man explains to Arya that there is in fact only one god, and all the others, men pray to, are just aspects of the one god.

 

[SiSt]

The dragons were traditionally named after gods, so we could see them fitting the patterns of inventing religions to fit the magic. 

Firmly in the "one magic, many religions" -camp. 

Why are the riverlands so important to the cotf? Because of the rivers? Seems many important places for the cotf are connected to water, so maybe its a way to make their magic flow? 

[Lost Melnibonean]

Apparently, my old theory that the power of Beric and Catelyn comes from the Old Gods, not the Lord of Light was just a crackpot full of mumbo jumbo. Here are the relevant portions of a recent interview with the George...

Quote

Q: You’re in unusual territory, with your characters very much still in your hands but also out in the world being interpreted for TV. Are you able to have walls in your mind such that your Daenerys, say, is your Daenerys, and Emilia Clarke’s Daenerys is hers and the show’s?

A: I’ve arrived at that point. The walls are up in my mind. I don’t know that I was necessarily there from the beginning. At some points, when David and Dan and I had discussions about what way we should go in, I would always favor sticking with the books, while they would favor making changes. I think one of the biggest ones would probably be when they made the decision not to bring Catelyn Stark back as Lady Stoneheart. That was probably the first major diversion of the show from the books and, you know, I argued against that, and David and Dan made that decision.

In my version of the story, Catelyn Stark is re-imbued with a kind of life and becomes this vengeful wight who galvanizes a group of people around her and is trying to exact her revenge on the riverlands. David and Dan made a decision not to go in that direction in their story, pursuing other threads. But both of them are equally valid, I think, because Catelyn Stark is a fictional character and she doesn’t exist. You can tell either story about her.

...

Q: Did Lady Stoneheart come about because it was hard to say a permanent goodbye to Catelyn?

A: Yeah, maybe. That may have been part of it. Part of it was also, it’s the dialogue that I was talking about. And here I’ve got to get back to Tolkien again. And I’m going to seem like I’m criticizing him, which I guess I am. It’s always bothered me that Gandalf comes back from the dead. The Red Wedding for me in Lord of the Rings is the mines of Moria, and when Gandalf falls — it’s a devastating moment! I didn’t see it coming at 13 years old, it just totally took me by surprise. Gandalf can’t die! He’s the guy that knows all of the things that are happening! He’s one of the main heroes here! Oh god, what are they going to do without Gandalf? Now it’s just the hobbits?! And Boromir, and Aragorn? Well, maybe Aragorn will do, but it’s just a huge moment. A huge emotional investment.

And then in the next book, he shows up again, and it was six months between the American publications of those books, which seemed like a million years to me. So all that time I thought Gandalf was dead, and now he’s back and now he’s Gandalf the White. And, ehh, he’s more or less the same as always, except he’s more powerful. It always felt a little bit like a cheat to me. And as I got older and considered it more, it also seemed to me that death doesn’t make you more powerful. That’s, in some ways, me talking to Tolkien in the dialogue, saying, “Yeah, if someone comes back from being dead, especially if they suffer a violent, traumatic death, they’re not going to come back as nice as ever." That’s what I was trying to do, and am still trying to do, with the Lady Stoneheart character.

Q: And Jon Snow, too, is drained by the experience of coming back from the dead on the show.

A: Right. And poor Beric Dondarrion, who was set up as the foreshadowing of all this, every time he’s a little less Beric. His memories are fading, he’s got all these scars, he’s becoming more and more physically hideous, because he’s not a living human being anymore. His heart isn’t beating, his blood isn’t flowing in his veins, he’s a wight, but a wight animated by fire instead of by ice, now we’re getting back to the whole fire and ice thing.

Beric, and then Catelyn, were reanimated by the last kiss of R'hollr. 

[aryagonnakill#2]

Good on you for correcting this yourself.  Most people on here would say George was talking about the show and just continue on.