In the first sentence, you are using the present simple for the first and the second clause. It means that the two actions happen in the present.
If it helps you, imagine those two actions happening in the same moment.
Although I don't know the context, it's more likely that the person involved with the crime did something before the rumour turns out to be true. So the first clause should be related more to the past than the present.
This is why I would use had.
Sam has already pointed out that the end of the sentence would sound better with turned out to be true.
So the outcome is:
The rumor that she had something to do with the crime turned out to be true.
Last thing, rumor is AmE, while rumour is BrE. Choose the one you prefer.