Before he sends Brienne off on her mission to find Sansa, Jaime gives Brienne a princely gift: a Valyrian steel sword remade from Ned’s great sword Ice. Through the magic that clings to Valyrian swords, the new sword, half the size of the old one, is also completely changed in appearance: Ned’s Ice is described as a smoky grey, while the sword Jaime ripples with the colors of “blood and black” (which are the Targaryen colors, by the way!) and the light on its edge is tinged red. (My personal theory is that somehow this sword is going to end up in Jon Snow's hands, because he - like the sword is a Stark/Targaryen mixture - and because he keeps thinking that Longclaw is the "wrong" Valyrian steel sword for him. But that's by the by ...)

It occurs to me that Jaime, like the sword he gives Brienne, is broken and then reforged (by suffering and by proximity to Brienne) into something different. Jaime goes into a captivity as Cersei’s double with his full complement of limbs and of Lannister arrogance; he comes out of it crippled, bearded, with silver in his hair, and with a resuscitated belief in the ideals he once held.

Oathkeeper is Jaime’s father’s gift to him, a bribe for Jaime to abandon his Kingsguard oaths and to do what Tywin wants. I find it fascinating that Jaime chooses a name for this sword that’s so unlike the other named Valyrian steel swords we encounter. 

They are usually named for some attribute of the House to which they belong, e.g. the Targaryens have a connection with dragons and their sword is Blackfyre; the Starks’ house motto is “Winter is coming” and they have Ice; the Mormonts who live on Bear Island have Longclaw, and the "golden lions" of Lannisters had Brightroar, now lost. Failing these house-connected names, the Valyrian swords we know about have sort of generic “strike fear into people’s hearts” kind of name like Heartsbane, Widow’s Wail, Red Rain or Lady Forlorn. 

But Jaime chooses a name that is quite different from either of these conventions: Oathkeeper (like Arthur Dayne's Dawn and Azor Ahai's Lightbringer) is a promise of better things to come, rather than a threat or a reflection of familial glory. When Jaime calls the sword he gives Brienne "Oathkeeper", he is indicating his changed intentions (to keep the vow he made to Lady Catelyn, when once he thought resentfully that vows made under duress weren't "real") and he is also rebuking his father, who wants him to break his oaths for the greater glory of the Lannisters. When Brienne, who witnessed Jaime broken and reforged, accepts that reforged sword named Oathkeeper from a man who intends to keep his vows, in a sense, she becomes his sword-hand, wielding Jaime’s sword to do what he physically cannot.

Brienne promises Jaime that she’ll find Sansa Stark “for her lady mother’s sake. And for yours” but she makes no solemn vows or oaths to Jaime and he asks for none, partly because Jaime, of all people, knows that vows are traps, but partly because how do you ask for a vow from a part of yourself?