Fingernails consist of keratin proteins, the same material used for producing hair. The most obvious purpose of fingernails, like hair, might be protection- protection of nail beds.
After reading the post, some of the hypothesis that I narrowed down to are:
1. Fingernails resemble animal claws that other species in the animal kingdom have. Perhaps they are vestigial remains of our wild past before the usage of tools.
2. Human species consumed a lot of fruits. It is widely possible that the fingernails helped us peel fruits.
3. Fingernails might have been useful for other daily tasks, such as scratching or picking away small items, such as the behavior of apes picking each others' fleas. A modern example of this usefulness is playing the classical guitar. This preciseness allows much more detailed movements.
A good leading point is to perhaps think about the evolutionary purposes of toes and toenails. Do/did they have similar functions with fingernails, and we just lost them once we became bipedal?
Bonus picture: longest fingernails in the world- this 45-year-old woman has been growing her fingernails for 18 years that are about 6 meters long now. I'm pretty sure she doesn't play the guitar. Or pick her nose.