What quality do top schools look for, besides normal requirements? In his answer in How do you get a bad transcript past Ph.D. admissions?, InquilineKea says:


  By the way, elite private schools (for whatever reason) tend not to have GPA cutoffs. If you're a student with a low GPA applying for an elite private school, you probably have something else in you that's extremely unusual, since very few students with low GPAs apply to them.


Sooo, what exactly are they? I guess, you are thinking this right now:


  Research ability. Number of papers. Period.


But that's nothing new for me (do I miss something?). I'm looking for the extremely unusual things that he mentioned. Googling around, I don't see much information given for grad students, only for the undergrad. In an answer in Quora, a women who self-claimed to used to interview applicants to MIT lists four qualities (again, for undergrad):


  
  Deep intellectual curiosity and a passion for learning that goes far beyond the norm.
  Self motivation, tenacity and grit.
  Lack of arrogance.
  A burning desire to attend MIT above all other schools, and can explain why.
  


Do those unusual qualities ever exist? How heavy are they in the final decision? And where should I represent them? In the emails to the professors, or in the SOP to the adcom, or just in my website? Or should I just stick to the research ability?
Graduate admissions is very different from undergraduate admissions. A lot of what you may read online about undergraduate admissions does not apply to graduate admissions.

(Traditional) applicants to undergraduate programs are understood to be (generally) young. They haven't had much of a chance to prove themselves yet through meaningful accomplishments. So in undergraduate admissions, it is not uncommon to look for intangible "qualities" that speak to a candidate's ability to be successful in areas that they haven't yet had a chance to prove themselves in. That's just because many highly qualified prospective undergraduate students don't have many tangible achievements (simply due to lack of life experience, not lack of potential.)

In contrast, applicants to graduate programs have more experience, whether academic, life, or professional experience. Prospective graduate students are simply not nearly as "unknown" as prospective undergraduate students. They have had more of a chance to collect some meaningful accomplishments. Thus in graduate admissions, tangible achievements weigh far more heavily. 

So while motivation, curiosity, etc. may be useful qualities for a graduate student, admissions committees are interested in what you have done (which may show that you have these qualities). They're not interested in having you tell them what qualities you believe you have.

Sometimes if a candidate lacks academic achievements but has meaningful achievements in other areas (e.g. entrepreneurial), these can help. (That may be what was implied by "you probably have something else in you that's extremely unusual".)