To have your say means to get an opportunity to deliver your opinion on something. If you're at a loud and hostile town council meeting, you might not get a chance to have your say unless you shout it.
Say means to speak, so any time you utter a word, you're saying it. If you write an editorial about dogs in the paper, that's also a form of saying. Someone could quote you as saying "dogs should be worshipped." You can say something with your body language, too. Like most common words, say is very old, coming from the Old English sęcgan, meaning "to tell."