In Star Wars: A New Hope when Kenobi saves Luke from the Sand People and wakes him up. Without Luke giving his name, Kenobi calls him,
Is there an explanation or is this just a movie mistake?
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Ben was present at Luke's birth and delivered him to his adopted parents at the end of Revenge of the Sith. He then installed himself in a nearby hovel to keep an eye on him and to defend him from potential attackers.
We see in the (canon) comics that he met with Luke on multiple occasions during his early years, initially visiting the family on occasion, then observing the boy from afar after his falling-out with Owen Lars, right up to age 17, just a couple of years before seeing him again in ANH.
Luke age (approx) 4
Star Wars #7
Luke age (approx) 7
Star Wars #15
Luke age 13
Star Wars Webcomic: #1 - An Old Friend
Luke age 17
Star Wars Webcomic: #2 - Meeting the Droids
"The droid says he belongs to someone named 'Obi-Wan'. Do you think he means Old Ben?"
To me, it seems clear from his preceding conversation with his Uncle that they all knew who Old Ben was, and so it reasons that on a desert planet with a tiny population that Old Ben would know who Luke Skywalker was.