The 3rd incarnation of Princess Zelda appeared in the SNES' The Legend of Zelda: A Link to the Past. While her role was generally the same as the original, I really liked this Zelda.
One reason of course is that you get to see her long before she is captured. You even get to save her at the beginning from a minor confrontation.
But then she gets locked up for good in the Turtle Rock dungeon.
So Link must defeat Trinexx to finally set her free. Which is a pretty sweet moment.