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.

In addition to what other answerers have said, you could use Latin-based anniversary names, such as centenary or centennial. Terms for other durations are less common, though.