Google Books initially claims about 12,800 results for "they have the say" - but if you follow that link and try to access the next page of results, GB admits there are actually only 31 (and less than a couple of dozen separate instances are relevant to OP's usage).
By contrast, "they have a say" initially claims 75,000 hits, which I expect are mostly both relevant and "real" (in fact, after I paged through some results, GB revised its estimate to 203,000 hits).
GB's algorithm for estimating results almost certainly pushed up the value for the first search term because it shares everything except the apparently trivial the/a switch with the second search term.
For completeness, "they have the final say" tops out at 82 hits (initial guestimate: 42,000).
So even though there's no grammatical argument against having the say, idiomatically it's clear that's a "non-standard" version.
In terms of a difference in meaning, if someone has "a say" in some decision-making process, it just means they're entitled to present their position in a discussion (they can say what they think). The implication of using the indefinite article is that other positions are also to be considered.
If someone has "the [final] say", it means they are the final authority. Whether or not they contribute to the debate itself, they will say what the final decision is (as, for example, a judge). The implication of the definite article is that there's only one participant who has this authority.
I think the two different senses for this noun usage of a/the say are nicely matched by...
"You will have your say later" (first meaning; [chance to] speak)
"It will only happen on your say so" (second meaning; declare/command, as final authority)