Villager who has a white cloud on his/her head, talk to them.

By the choice of the conversation, you can ask them not to leave.

The visitor will pick one resident randomly for trade, when your island is full.

If the visitor chooses the resident you want him or her to stay.

You can trade the visitor with the unwanted resident, by using Amiibo Card.

In order to trigger this event, you need to listen to the visitor's request 3 times.