There isn't any way to prioritize a certain villager. Some people swear by decreasing or increasing friendship with the villager you want to move out but mostly it seems to be random.
However, there's a TT method I use that makes it relatively easy to get a specific villager to move. It's never taken me very long to do that. It does affect your town since it makes weeds appear and makes your mayor get bed hair.
To start with, you find a week that has no events coming up and check which day is the Sunday that begins that week. Make sure you've denied a villagers request to move out on the day when you'll be attempting this, so if you're trying this today tell a villager to not move and only TT once you've done that. I don't know if the Sunday thing is absolutely required, the guides I've seen have mentioned it but it may be so that you won't accidentally travel into the week after that if there's an event in it.
Use the 3DS calendar to set it to that day. Go to the game, load as your mayor, save and quit. Go back to the 3DS calendar and set it forward by one year, so if your date is January 1 2014, set it to January 1 2015. Go to your game, start the game as a new character. Once you get the town map from Porter, check if anyone is missing from the map. They probably will be. Quit without saving. Go back to the calendar, set the date back by one year again, load the game as your mayor. Go find the villager who was missing and see if they ping you. If they don't, talk with them a few times to see if they mention it. If not, save and quit, then reload the game (without adjusting the calendar) and try again until they ask you to move.
If it's someone you want to keep, tell them to stay, save and quit. Then go to the calendar, set it forward by one year again. Go to your game, load as your mayor, save and quit without doing anything else. Go back to the calendar, advance by one year, then start as a new character. Check if anyone is gone, quit without saving, go back one year and talk to them with your mayor, etc. Basically, repeat the steps above.
Once it's a villager you want to move out, tell them to go, save and quit. Now you can TT back to your original date. Just go to calendar and set it. Just keep in mind on this step that TTing backwards any amount of time counts as one day forward, so if your villager was moving in 5 days they'll now move out in 4 (in case you need to calculate when they'll be in boxes for a trade).
Example-let's say you're starting the game today, July 3 2014. I don't know if there are any events next week but let's assume there aren't. Refuse a villager's request to move out. Use the 3DS clock to set it to next sunday, July 6 2014. You start the game as your mayor, save and quit. Then you set the calendar to July 6 2015 and start up a new character as described above. You go back to July 6 2014 to ask the character to stay. Then you set the calendar to June 6 2015 again, but this time you load the game as your mayor and save and quit. Then you set the calendar to July 6 2016, start up as a new character, quit without saving, then load with your mayor on July 6 2015. And so on until the villager you get the villager you want to move out. Finally, if you were for example on July 6 2019, you can TT back to July 3 2014. That counts as one day forward so if your villager said they'd be moving on July 11 2019, they'll now move on July 7 2014.
I hope that helps. I know some people use this but TT using smaller time intervals but I don't know how that'd work event-wise so I just go by year. There are also other variants on this method. Find faerie's guide on it in the forum, that's how I learnt about it and it lists the other variants too.
you just have to be lucky. I TT'ed all the way to my BIRTHDAY (November 9th) to move out Flora. She wouldn't budge until I returned to my main time and a few days later, she asked to move.
