I am very interested in the work of a professor who works for department X. His work is interdisciplinary and I am interested in the engineering aspects of his work. I contacted him and he agreed to be my primary mentor as long as I had a secondary mentor through engineering school since I am not going to apply in his department. He mentioned a specific name in the engineering school who shares the same research interests. Is it a good idea to mention his name along with the name of the professor in the engineering school in my Statement of Purpose (SOP)? For instance, that I would like to join the engineering lab and conduct a collaboration with the other lab? Is it a good idea to show an interest in a lab who is not in the department but still related to engineering?
I think that in general this would be fine, but don't give the impression that working with him or the other lab/prof is, in any way, a condition of your acceptance. You want the department that you are applying to to be interested in you but not to give others the impression that you are inflexible in any way. But stating that you have a well worked out plan and have at least partial buy-in on it from a professor shouldn't hurt you, and might help.
Your first goal is to get accepted, then work out the details. Make it obvious that your background supports your SOP goals.