Why did Aro kill Diddyme?

Because her and Marcos were so happy together that they were going to leave the Volturi and Aro would not allow that..so he killed his sister so that Marcos wouldnt leave.

Aro killed Didyme because she was jeopardizing his empire. After Marcus fell in love with Didyme, he lost interest in helping the Volturi advance, which did not bode well with Aro. So Aro killed Didyme in order to ensure that Marcus would remain with the Volturi. I don't believe this is mentioned in the book, but it can be found in the Question and Answer portion of Stephenie Meyer's website.