This page provides information on how to deal with a sick villager and also increase your friendship rating with said villager.
Finding them sick in their home
When you first come across a sick villager, you’ll see them in their pyjamas shivering and sneezing. You should speak to them right away, as ignoring a villager in their home is rude and will negatively impact your friendship. When you talk to them, they’ll mention that they need medicine.
Giving them medicine
Medicine can be made with 3 weeds and a wasp nest, or you can buy some at Nook’s Cranny. You should do this straight away, as your villager will not get better the next day and you’ll negatively impact your friendship since you’re ignoring his need for help. If you also say you have medicine and then you don’t actually give them any, this will also impact your friendship in a bad way.
When you give them medicine, your villager will thank you by giving you a gift.