Why does the same person with no mutual friends or connections always rank high (or on top) every single day in the "People You May Know" list on Facebook? How does the ranking work?
Only Facebook knows for sure. I'm sure their algorithm is just one of many trade secrets.
They do say in Facebook Help:
We show you people based on mutual friends, work and education information, networks you’re part of, contacts you’ve imported and many other factors.
Since you say there are no mutual friends between you, perhaps they went to the same school as you, or worked in the same company, or live in the same town. Maybe you and a bunch of your friends happen to have them in your contacts.
Facebook bought a company called Octazen back in 2010, whose specialty was scraping user data. There could be any of a number of connections between you that you don't realize (according to Facebook) and everyone else who would "rank higher" is probably already your friend (or you've blocked and won't show in the list anyway).
But, really, without a Facebook engineer who knows the algorithm to tell us, we can only guess.
One factor not mentioned in earlier answers is that you and this person might have been geographically close to each other many times, e.g., you might play tennis at the same location at the same time, commuted on the same train every day or something like that, even though you never talked to each other. This might be enough for Facebook to connect you to each other.
I read an article about how Facebook use geotracking to connect people a while ago describing this. Unfortunately I can't find it again so I can't provide a source.