Okay, so my son and I just finished our first game. In the final turn, I found his base, and tried to take the system. He destroyed all of my ships in space leaving him with a handful of fighters. I destroyed all of his ground units. After reading and re-reading the win conditions, it appeared that since he still had ships in orbit, the empire didn't win. He then brought in extra units with rapid mobilization, my remaining ground units were destroyed and the rebels won the game...

So, did we play this right? Even though all those pesky rebels were destroyed on the planet surface, the empire didn't win it?

What you describe is what happened in Hoth, the Empire crushed the rebellion on the ground, but the rebel ships escaped so they lived to fight another day.
You interpretation on that was right, to win there must be no Rebel units of any kind in the rebel base system.

What you did wrong if I understood well is that the Rapid Mobilization card doesn't let you move units to the Rebel base once it is revealed (re-read the card when you have the chance and you will see that it clearly states that).

Got it, thank you!

Was expecting a Charlie Sheen joke. I am disappoint

Sorry to disappoint...