It is supposed to be Ser Arthur Dayne. The actor Luke Roberts is credited as Ser Arthur Dayne. The other actor is Eddie Eyre, credited as Gerold Hightower.
Furthermore there is some revelation in the episode which was poorly executed, hence causing all the confusion. Bran did note it was Arthur Dayne but somehow it was not clearly aimed at the man.
Bran: That's my father.
Three eyed Raven: The man beside him is Howland Reed, Meera's father.
Bran: Ser Arthur Dayne. The Sword of the Morning. Father said he was the best swordsman he ever saw.
Then he proceeds to say the iconic line Ser Arthur Dayne said in the books:
Arthur Dayne : Lord Stark.
Young Ned : I looked for you on the Trident.
Arthur Dayne : We weren't there.
Gerold Hightower : Your friend the Usurper would lie beneath the
ground if we had been.
Young Ned : The Mad King is dead, Rhaegar lies beneath the ground. Why
weren't you there to protect your prince?
Arthur Dayne : Our prince wanted us here.
Young Ned : Where's my sister?
Arthur Dayne : I wish you good fortune in the wars to come.
[Arthur and Gerold don their helms]
Arthur Dayne : And now it begins.
[the two Kingsguard draw their swords]
Young Ned : No, now it ends.
There is more evidence when Bran notes his father getting hammered at the hands of the Sword of the Morning.
Bran: He's better than my father.
Three Eyed Raven: Far better.
Bran: But Father beat him.
Three Eyed Raven: Did he?
Bran: I know he did. Heard the story a thousand times.
We know here who exactly did his father beat. He was beaten when Howland Reed stabbed him the back, otherwise Bran's father would have died there.
But of course, it wasn't really the same as the books. In the books, Ned never talked about that to his children. Meera and Jojen were surprised to learn that Bran did not know about the story of Harrenhal either. He never talked about Ashara Dayne, his supposed lover, either as Arya learned from Edric Dayne. Dawn was not shown in the show at all, I guess because of CGI costs as Dawn is supposed to be milky white. They could have used enamel but I don't really know why they chose not to. And of course, Ser Hightower is a lot younger in the show than he actually was. They gave the White-Bull's lines to Arthur Dayne, I guess mostly because he means nothing to the show audience, they don't know him and he is not supposed to leave a lasting impression on them unlike Ser Arthur Dayne.
In the books, this is how it happened:
“I looked for you on the Trident,” Ned said to them.
“We were not there,” Ser Gerold answered. “Woe to the Usurper if we
had been,” said Ser Oswell.
“When King's Landing fell, Ser Jaime slew your king with a golden
sword, and I wondered where you were.”
“Far away,” Ser Gerold said, “or Aerys would yet sit the Iron Throne,
and our false brother would burn in seven hells.”
“I came down on Storm's End to lift the siege,” Ned told them, and the
Lords Tyrell and Redwyne dipped their banners, and all their knights
bent the knee to pledge us fealty. I was certain you would be among
“Our knees do not bend easily,” said Ser Arthur Dayne.
“Ser Willem Darry is fled to Dragonstone, with your queen and Prince
Viserys. I thought you might have sailed with him.”
“Ser Willem is a good man and true,” said Ser Oswell.
“But not of the Kingsguard,” Ser Gerold pointed out. “The Kingsguard
does not flee.”
“Then or now,” said Ser Arthur. He donned his helm.
“We swore a vow,” explained old Ser Gerold.
Ned’s wraiths moved up beside him, with shadow swords in hand. They
were seven against three.
“And now it begins,” said Ser Arthur Dayne, the Sword of the Morning. He unsheathed Dawn and held it with both hands. The blade was
pale as milkglass, alive with light.
“No,” Ned said with sadness in his voice. “Now it ends.”
So Lord Commander Gerold Hightower, Ser Oswell Whent and Ser Arthur Dayne were present there. And of course, Arthur was wielding his two handed greatsword Dawn.
Show depicted only Ser Hightower and Ser Dayne. In the books as you can see, Ser Gerold was the first to answer and did most of the talking, as befit his rank as Lord Commander of the White Swords.