What an interesting question! You will probably get a lot of varied responses to this question because I can think of a lot of plays myself that I would like change as far as the ending. That said, however, I believe that all of Shakespeare's plays are brilliant and the endings, although not satisfying to us, are nevertheless appropriate because they are tragedies, or comedies, or histories -- and Shakespeare's brilliance should not be messed with.
So, I answer purely on the basis of storyline. I think I would change the ending of Romeo and Juliet. The reason I choose this one is because Romeo and Juliet are teens. What happens to them is beyond their control and caused by the adults in their lives. Their love is pure and magical, and it always seemed unfair to me that the innocent should die, and not the guilty. Of course, that is how life is, but wouldn't it be nice if instead of them both dying at the end, Friar Lawrence appears with an antidote to their deaths? In my version, the Montagues and Capulets would hold a joint funeral for the lovers. Friar Lawrence would give the eulogies, chastising the parents for their stupid feud. When they were all feeling guilty enough, Friar Lawrence would sprinkle the antidote on the bodies and they would come back to life and live happily ever after.
The play would not be as powerful with this ending, but it is my fantasy.
What do you think?