If they are lucky, some new trait will appear that might be beneficial. But it will be subtle and no one will notice it. The process that creates a new potentially useful trait is unlikely to create that same trait in more than one person. It will be many generations until the effect becomes common, not until most people that don't have the trait have died offmost people that don't have the trait have died off. The vast majority of people will be descendants of that one individual.
There is a problem though. New mutations can take a long time to spread through the population, and meanwhile something must kill off everyone except for the one person with it and his descendants. Anything that will kill off most people within a few hundred years will already be killing them now. It will almost certainly be too late for any new beneficial mutations to spread through the population before the population becomes extinct.
Natural selection doesn't work well in a stable society. Without the sudden massive killing of large parts of the population, it isn't going to happen.
- ##Selection: As long as society is already handling the deformity problem, perhaps they can expand the definition of deformity to include some of the more obvious weaknesses. For instance, should children with weak eyesight be allowed to survive? If not killed outright, perhaps they could be sterilized and used as slaves.
As noted, natural selection isn't going to work in a stable society. That society is going to have to implement an unnatural selection process (eugenics).
Society is already handling the deformity problem, so perhaps they can expand the definition of deformity to include some of the more obvious weaknesses. For instance, should children with weak eyesight be allowed to survive? If not killed outright, perhaps they could be sterilized and used as slaves. Maybe remove people with short arms or fingers? Or if your world is very sunny, then get rid of people that don't have dark skin.
It will be up to you, the author, to decide what the world is like and what attributes would help or hinder society.