My mother had a stroke and now has a continuously runny nose.

Wow, my mom has a continuous runny nose and she is on Aricept. Thank you for that. I did not know.

I guess that at first her mother needs an doctor especialized ear nose and throat (ENT doctor) and to a doctor espexialized in allergy.
But if her mother it is taking acetyl cholinesterase inhibitora as donezepil (Aricept,etc.), that drugs are a common cause of runny noses.(it is a side effect of acetyl cholinesterase inhibitors).

I'm with Carol. I don't see how a neurological condition such as a stroke could cause inflammation of the mucous membranes which would cause the runny nose. Perhaps she has a chronic sinus infection or allergies. Why don't you take her into the doctor and get it checked out?

The only thing you can really do is consult the doctor. If you aren't satisfied with the doctor, try to get a second opinion. The runny nose may have something to do with her stroke or it could be a secondary condition that can be helped, so please try.
Good luck,
Carol