It’s an iconic headband we see appear all throughout the series, but why does Sasuke give Boruto his headband?
Sasuke chooses to give Boruto his headband because he accepts him as a shinobi. Boruto was recently dishonored when Naruto took his headband away after a cheating incident at the Chunin Exams. Sasuke gives Boruto his headband when the boy agrees to save Naruto.
After taking the headband, Boruto and Sasuke teamed up with the Five Kage and set off to save Naruto from Momoshiki.
Sasuke gives Boruto his old headband
In chapter #6 of Boruto: Naruto Next Generations, Boruto is looking around his father’s Hokage office. Naruto was captured by Momoshiki, and Boruto feels especially upset because he disappointed himself in Naruto’s eyes earlier.
Boruto had to fail the Chunin Exam because he used scientific ninja tools. This resulted in Naruto disqualifying his own son and taking his ninja headband away from him as a sign of disgrace.
Then, Sasuke enters the Hokage’s office. He alerts Boruto that he can sense Naruto’s chakra as a sign that he’s still alive, although the two of them should hurry. Sasuke assures Boruto that besides the boy’s recent disappointments, he still considers Boruto his student. Considering he’s Naruto’s son and trained by Sasuke, Boruto acknowledges his chances of long term success are great.
Sasuke compliments Boruto in his newfound interest in learning about Naruto’s successes instead of his failures. The other Kage arrives to back the two of them up. Before they leave, Sasuke gives Boruto his old headband.
He gave Boruto the headband out of respect. The two came to a mutual understanding in their Hokage-office conversation that Boruto would never cheat his way to success again. Sasuke completely accepts Boruto at this moment and grants him his head band.
Considering Naruto took Boruto’s headband away, Sasuke giving his up could symbolize forgiveness and a re-accepting back into the Leaf Shinobi.
So that’s the answer. Sasuke chooses to give Boruto his headband out of respect. Boruto admits he was wrong for cheating in the Chunin Exams. Sasuke gives him his old headband after the two have a talk about it.