"<Old name> ... I know that's your real name."
  "No, I'm Alyssa."
  "Honey, that's not her name anymore. Sometimes when people go through big changes in their life, they get a new name. Before I got married to Daddy, I was <Name Maiden name>."
  "I want my name to be 'Spider-Man!'"
  "Okay, Spider-Man, get in your car seat now."

That's awesome. And from that time, I would indeed start every conversation as "Hey there, Spider-Man, *words words words words...*"

Hee hee... Have to love kids' takes on this stuff.

When I told my 5 year old daughter I was becoming a woman, and my name was going to be Diana, she was all for it. But she insisted her name was now "Plum." She even had her classmates at school AND her teacher calling her "Plum." Lasted for about a month. Adorable. But... she likes calling me Diana still more than either of the other kids.