During the second episode of Game of Thrones, the camera took a lot of time to focus on Jaime’s sword as it was returned to him after his “trial” in the Great Hall of Winterfell. That must mean there’s something important about the sword, and it’s true. There is.
Yes, Jaime’s sword is made of Valyrian steel. He did give his first Valyrian steel sword to Brienne, but now he has another one.
Oathkeeper, Jaime’s original sword, was one of two Valyrian steel swords that were “re-forged” after Ned Stark’s Valerian sword Ice was melted down by Tywin. Brienne named the sword Oathkeeper.
Oathkeeper was one of two Valyrian swords made from Ice. Ice was originally an heirloom of House Stark and Ned’s sword. Ned used Ice in Season 1 to behead a deserter, and Ice was later used to behead Ned.
Then, Tywin had Ice melted down and the sword was made into two smaller Valyrian steel swords. The other half of Ice was given to Joffrey, who named it Widow’s Wail.
Tywin gave Oathkeeper to Jamie. After Joffrey’s death, Jamie gave Oathkeeper to Brienne with the task of getting Sansa somewhere that was safe and far away from Cersei. Giving her Oathkeeper was also symbolic, as he was returning part of Ice to the Stark family. Brienne is the one who named the sword Oathkeeper.
Oathkeeper has a golden lion on the hilt, which gives away that it’s now of Lannister origins. Brienne has used Oathkeeper to protect Sansa and to win a battle against Sandor. She used it to rescue Sansa and Theon at the beginning of the season. Brienne offered to return the sword to Jamie. But he refused to take it back.
Instead, Jaime now has the Valerian sword called Widow’s Wail, which is the other half of Ice. The sword was one of two forged from Ned Stark’s sword, Ice, after Ice was melted down. It was given to Joffrey, who obnoxiously named it Widow’s Wail.
Jaime later obtained Widow’s Wail himself and he was spotted wearing it in multiple scenes.
Widow’s Wail was passed down to Tommen, but Tommen never took the weapon. In Season 7, Jaime began wearing Widow’s Wail after he returned from the Riverlands. Olenna even noticed Jaime wearing the sword and asked if he was going to kill her with it. Jaime tells her the sword is Widow’s Wail.
So yes, the Valyrian sword being returned to Jaime was very significant. Some people think that Widow’s Wail and Oathbreaker will ultimately be reforged to create Ice, which might be the legendary Lightbringer sword wielded by Azor Ahai. This feeds into the theory that Jaime is Azor Ahai and Brienne, perhaps, is his one true love. (Considering how Azor Ahai’s story went, let’s hope that’s not the case.)