None of them bother me;
She opened the mail which was delivered that morning.= The mail that was delivered that morning.
She opened the mail which had been delivered that morning.= The mail that had been delivered that morning.
I agree with cyberpedant, who illustrates the value of pedantism*.
To answer the question, I think the past perfect is commoner.
Both sentences are perfectly understandable and I think correct but the past perfect is better because it makes it unambiguous that the letter was delivered before she opened it. "that was delivered that morning" sounds vaguer and I feel like putting "that same morning" to make it more precise. I would say use past perfect.
"pedantism" ??? Tsk, tsk and a finger-wave to you, Gorgiewave. It's "pedantry." And I ought to know!
Akasaka asks: "Understandable, but when you see or hear the first sentence, it bothers you?"
Not really, but if I were grading someone's paper, I would underline it.