I think you just have to talk to her a lot and reach that 'best friend' zone, because usually villagers are most likely to move out when they've reached the highest friendship level and whatnot.
Or you can do that 2 days forward, 2 days back and just wait until she asks to move out. That might work, also.
I think there's this thing where the newest person in your town won't be the next person to move.
By the way, if you have 8 villagers or less, expect a new one.
If you have 9 or 10, someone'll want to move out in 4-5 days.
The most recently moved-in villager can't ping or move out, so somebody has to move in after her. Make sure you talk to her so she can ping you, but you don't have to be "close" with her.
I suggest talking to her at least once a day to make sure she doesn't do the "We haven't talked in a while" thing, otherwise she won't ping you.
I know a lot of people have already chimed in, but the best way that I've found is to talk to the person the FIRST day they're boxes moving in. Introduce yourself, and then completely ignore their existance forever. Eventually they'll feel unwanted and move out. You can exhaust a chat with a villager to the point of him/her telling you whether a villager is thinking about moving out. I do not do this, but for some people it works.
I've done this twice; each villager took about about a week to move out (which isn't really that long).
While I was cycling, I had to move a bunch of villagers in and out. For me, I find it easiest to move out the newest villager by having one move in and another move out. They won't move out otherwise. But good luck!
