“The next stop will be...”

You're correct: the next stop *is* X, because the order of stops is a predetermined list.

But they are also correct: the next stop *will be* X, because you haven't stopped there yet.

I have a question. Should it be "next stop" or "THE next stop"?

A case could be made for either construction, but I think context trumps syntax in this instance. That context is the subway. The phrase "the next stop will be" is more likely to be understood in a crowded train than "the next stop is." The sibilant "is" would be more easily drowned out than the sharper-sounding "be."

While travelling by train, although the train will be stopping at the next station shortly, it is the next stop. All later stations the train stops at on the journey will be the next stop at some time in the future but the next stop the train will be stopping at applies now.
The announcement is typical “The next stop will be...”.

I prefer “The next stop is” as this is definitive and applies to the next stop; whereas “The next stop will be” applies to all future stops and is not referring specifically to the next stop, which in most cases it is.