When a whiny farm boy (Luke) shows up at the secret base of the Rebel Alliance, a group of high profile exiles and crack veteran warriors, he is instantly included in their fighting force, even taking a senior role. In movies 4-6, no one (apart from Obi-Wan) on the Rebel side ever comments about Luke's last name or makes any connection to Anakin Skywalker, who presumably was a famous heroic figure among the former Republic Illuminati running the Alliance (remembering that it wasn't well known that Anakin became Vader, he was presumed killed along with the rest of the Jedi).
Did they realise Luke was Anakin's kid? Maybe that helped in getting Luke involved in the fight and promoted so quickly. Or did they not know, or simply not think to connect the names? Maybe 'Skywalker' is like 'Smith' on Tatooine (although it seems pretty grandiose, Anakin's slave mother bore the name)? Maybe there's lots of teenage 'Skywalkers' running around having been named in honour of the famous celebrity from the Clone Wars, so the Alliance honchos just assume Luke's parents were just another couple of hillbillies giving their kid a stupid fashionable name?