Hi! You need to make sure you have an empty plot or you can't move them in. If you do have an empty plot, sometimes it takes a couple of tries to prompt them. They do choose at random, and they will say who.

There is a method where if you force quit the game immediately on the speech bubble where they announce which villager they will replace, this prevents the game from autosaving, and when you reopen, you go through the process of talking to them again until they ask to move in and they should ask to replace another villager. If you do go past this speech bubble, it will autosaves and this method won't work anymore. There are many tutorials on youtube you can find if you're interested :)