Cognomens were third names typically adopted in adulthood by citizens of ancient Rome. This was generally a nickname. According to some ancient Rome wiki "Cognomina often, but not always, referred to a person's appearance or other characteristics". These usually fall into one of several broad categories such as Occupational, Adoptive, Geographical, or Honorific.

They also have a list of some of the most common Cognomens, I will give you the highlights. Note: Each Cognomen typically has a male and a female form. I will just be listing the male form for simplicity.

Cognomen Meaning
Blaesus Someone who frequently mispronounces words or has a lisp
Caninus Someone who looks like a dog
Cilo Someone with large lips
Habitus Someone in good physical condition
Postumus Someone born after their fathers death

Usually a Cognomen was neither self assigned nor assigned by ones parents. It was either inherited from his parents or given by general consensus within the entire community. So say I had very small eyes, small enough to be some distinguishing characteristic, and I have no other particular claim to a greater title, my neighborhood might decide that I am now known as "Lucas Wilson Ocella", where Ocella was typically given to people with small eyes.

I actually think this is pretty fun, but it seems to be a practice thoroughly lost to time. I cant find any modern culture that practices anything similar.

I came to learn about this topic from the ever-interesting Tor's Cabinet of Curiosities video on Myths and Mysteries of the Early Christians. Its great and I recommend you check it out.