If they don't choose the right villager, you'll need to re-roll by completely exiting the software and going through the process again. You can accept their initial offer if they're trying to trade places with the critter you dislike. This is key: They'll randomly choose a villager to evict.
If you have a full island with 10 residents and a villager visiting your campsite, you can talk to the visitor over and over again to convince them to move to your island.
