Should one mention the prestige of the target university/advisor as motivation in a statement of purpose? When one writes their statement of purpose, is it advisable to mention the prestige of the target university?
How about expressing interest in working with a certain professor due to him being world renowned in the student's field of interest?

On the one hand, it seems quite common, but on the other hand I'd think the committee would get the impression that the student is applying randomly to top universities (and/or to work with famous researchers), instead of choosing based on specific research interests.

Thanks!
When applying for a graduate degree, the prestige or other characteristics of the university as a whole are generally irrelevant, compared with the specific department or graduate program.  I would strongly recommend against mentioning the university's prestige at all.  (You definitely don't want to give the impression that you see being able to list Harvard on your CV as comparable in importance to intellectual issues.)

It's common to discuss characteristics of the department or potential advisor, but you shouldn't focus on fame or prestige.  Instead, you should highlight the more fundamental aspects that matter to you.  There's no point in saying something vague like "I hope to attend your graduate program because everyone agrees it's great."  (Assuming this is not just empty flattery, the admissions committee already knows they have a great reputation, and it's irrelevant for your admissions decision.)  Instead, you could say something like "I hope to attend your graduate program because of the opportunity to work with leading experts in X and Y."  This might be highly distinctive if few departments have strength in both X and Y, and even if this is common it's still more substantive than just talking about prestige.