Jaime’s Golden Arm is Symbolic of Brienne and the Better Person She Helped Him Become
Jaime lost his sword hand as a result of his actions (negotiating) to save Brienne from the Bolton men. The first act of compassion and selflessness we really see him perform.
Then when he hands Brienne Oathkeeper for the first time, it is passed on with said missing hand.
Now, when he waves goodbye to her at Riverrun he also uses that same arm.
Essentially, Brienne represents that missing sword hand. He sacrificed it to save her and that part of him will forever be changed. It is symbolic of everything they have endured together and it is something to remind him of how she brings out the best qualities within himself.
Jaime was ready to simply give up and die due to the loss of his best quality. But Brienne gave him the will to live and to face adversity. Throughout the rest of their journey she taught him that he was worth more than a mere sword hand.