Game of Thrones
As per show canon we also don’t know. IMDB refers to him as “Red Keep Stableboy” so that is probably the characters official name.
The wikia on him also just refers to him as “Stableboy”.
Westeros.org also refers to the character as just "Stableboy".
Lastly, the scene happens so quickly so we never get the chance to learn his name. Here’s what the transcript has to say, again referring to him as “Stable boy”.
Arya Stark: Needle.
A stable boy appears.
Stable Boy: There she is.
Arya Stark: What do you want?
Stable Boy: I want you, wolf girl. Come here.
Arya Stark: Leave me be. My father's a lord. He'll reward you.
Stable Boy: She'll reward me, the Queen!
Arya Stark: Stay away!
Accidentally, she turns around sliding Needle through the stable boys belly. Her first kill is a fact.
Shook as she is, she runs off.
Game of Thrones, Season 1 Episode 8, “The Pointy End”
A Song of Ice and Fire
Again per book canon we don't know, when the stableboy has a chance to introduce himself he declines.
"There she is," a voice hissed close behind her.
Startled, Arya whirled. A stableboy stood behind her, a smirk on his face, his filthy white undertunic peeking out from beneath a soiled jerkin. His boots were covered with manure, and he had a pitchfork in one hand. "Who are you?" she asked.
"She don't know me," he said, "but I knows her, oh, yes. The wolf girl."
A Game of Thrones, Arya IV
Every time she remembers the event later on she refers to him as "Stableboy" too, for example.
Arya was remembering the stableboy at King's Landing. After him there'd been that guard whose throat she cut at Harrenhal, and Ser Amory's men at that holdfast by the lake. She didn't know if Weese and Chiswyck counted, or the ones who'd died on account of the weasel soup . . . all of a sudden, she felt very sad. "My father was called Ned too," she said.
A Storm of Swords, Arya VIII