Depending on your preference:
snot (informal), nasal mucus (formal) or nasal discharge (medical-sounding)
The correct medical term for runny nose is
Rhinorrhea or rhinorrhoea is a condition where the nasal cavity is filled with a significant amount of mucus fluid. The condition, commonly known as a runny nose, occurs relatively frequently. Rhinorrhea is a common symptom of allergies (hay fever) or certain diseases, such as the common cold. It can be a side effect of crying, exposure to cold temperatures, cocaine abuse or withdrawal, such as from opioids like methadone. Treatment for rhinorrhea is not usually necessary, but there are a number of medical treatments and preventive techniques available.
Mucus is something everyone has, and some people wish they had a lot less of the stringy, gooey stuff. Sure, it can be gross to blow globs of snot into tissue after tissue when you have a cold or sinus infection, but mucus actually serves a very important purpose. […] during an allergic response to an offending trigger, such as pollen or ragweed, most cells in your body squeeze out a substance called histamine, which triggers sneezing, itching, and nasal stuffiness. The tissue of the mucus membranes starts leaking fluid, and your nose begins to run.