The weather forecast says it will rain tomorrow.

1 would rain is the past tense and the future tense, and the usage of would is only consistent with the main sentence predicate verb said.

Either way, it all depends on the speaker's mood.

If he emphasizes that the weather forecast is the past, use the first one;

If we take the weather forecast as a fact, then we can use says.

To put it bluntly, either sentence is ok, depending on the mood.