Is it correct to use the word "birthday" for the deceased, or is there a better alternative?

I would say "Today would have been ...", especially since you are talking about a specific number (80). Because as a living person he didn't turn 80, so he never had that birthday.

But if you leave out the number I think you could say "Today is my uncle's birthday".

As masarah pointed out your second form is correct, but there is also a correct way of wording your first form in American English. It would be:

Today is the 80th anniversary of my uncle's birth.

In other words we generally don't consider it somebodies birthday any more (except with a "would have been" qualifier), but particularly for notable figures in history we still count the anniversary of their birth.