Here's an interesting thing about the "Skywalker" name.

Luke never actually uses it to describe himself until he meets Leia and says "I'm Luke Skywalker. I'm here to rescue you."

Now, I know this may be contradicted in some of the novels and other EU shenanigans, but it's possible (if we go off the movies alone) that Luke lived under the name of "Luke Lars" until he learnt about his father from Obi-Wan. After learning that his father was a Jedi and after realising he would have to leave his life on Tatooine behind, he may have decided "Hey, I want to be a Jedi. I want adventure and excitement - I'm going to re-adopt my father's name! I'm going to re-create myself! I'm going to let the universe know I'm the son of Skywalker!"

Obi-Wan might have advised him against this, but at this point he was out of the picture and couldn't stop him. Then Luke becomes a famous hero of the rebellion, and Vader's interest is piqued.

Edit:

On an aside, if any of these ideas were to be actually filmed I would love for it to be done using exclusively the special effects techniques available at the time the original trilogy was made.