Ignoring a character won't work and making a character from a different town become best friends with the villager you want to move out won't work either. If anything, only what you do with the mayor of the town you're trying to move a villager out of would count but all the friendship methods are a bit hit-or-miss because it's mostly random.
The easiest way I know to move out a villager is the one I do. It sounds a bit complicated but I got it from a thread here on TBT that had some variations, and it's easy once you get the hang of it. This involves TTing and you'll need a spare spot of a new player character (that will be deleted later). All of this is about your brother's town by the way.
In my example, I'm going to use the time I used this method that sticks out to me the most, which was in June of this year. You don't need to use that date, it's just an example. I'll be putting my example in brackets (like this).
The first thing you have to do is choose a week with no upcoming events and find out what the Sunday that starts that week in the game is. In my example, I was in late May and the date I picked was June 1 2014.
Next, set your 3DS clock to that date you chose. If you're going ahead more than one day, maybe two at most, don't attempt this method right away. Just wait until a villager pings you and (unless it's Merengue) ask the villager to stay, and only then TT ahead. This is because when you skip ahead, you risk losing villagers, but if you deny a villager's request to move before skipping nobody will move out.
When you're ready to start, load the game as your mayor. You should now be on the chosen date (in my example, that would be June 1 2014). Save and quit. Go back to your 3DS settings and set the date one year ahead (for example, June 1 2015 instead of June 1 2014).
Start the game as a new character. The answers you give to Rover don't matter because you won't be keeping this character. When you arrive at the train station, Porter will give you the map and tell you to go to the town hall. Don't go to the town hall at all during this. Just check the map very carefully and see if anyone has moved. If you want to be extra safe and nobody is missing from your map, go to the houses of villagers who are indoors and see if any are in boxes.
Quit without saving and go back to the 3DS calendar. Set it to one year before again (June 1 2014). Load the game as your mayor and go talk to the character who was missing from your future map. If they don't ping you to move, talk to them some 5-6 times. If they say nothing about moving, save, quit, reload the game (still on that date) and go find them again. Repeat this step until they do ask you to move.
If it's a villager you want to keep, ask them to stay. Save your game and quit. Go to the 3DS clock and set it one year ahead (June 1 2015). This time, when you start the game, load as your mayor, save and quit. Then go to the 3DS clock and set it another year ahead (June 1 2016). Use a new character to check if anyone is missing. Go back one year again (June 1 2015) and use your mayor to go talk to the villager who is missing in the future as I explained above. Keep doing this.
When Merengue is the one asking you to move, tell her to go. Save the game and quit. Then go to the 3DS clock and set it back to your original date. TTing back by any amount of time equals one day forward in game so you'll need to take this into account when calculating the new moving date. For example, if Merengue said she'd leave 5 days later, she will now be leaving 4 days later. In my example, if she said she'd be moving on June 6 2025, and your original date happened to be May 29 2014, she would now be leaving 4 days after that.
The downsides to this method are that you'll get weeds and bed hair (you can fix the latter at shampoodle). It will also mess up your activity log and if any stores are due to open in your town, when you check your town records they may show you a random other year but I personally think that's bearable. There are methods that make this less awkward by going week by week or month by month but I find year by year less confusing, especially since events can accidentally throw you off because characters won't ping then. Don't forget that if you and your brother share the same 3DS, you should change the 3DS's clock back to your normal date before playing your game if he interrupts this process at some point and then change it back when he starts again.
- - - Post Merge - - -
I should add, if your brother doesn't want to TT, all you can do is be patient and let him keep talking to Merengue regularly until she offers to move out. The 10th spot in your town won't be filled at random so you can save it for Merengue. That spot can only be filled by villagers you invite (from other towns or the campsite), and I think it can also be filled by random villagers you may get from wifi or Streetpass (but you can avoid that by disabling Streetpass and not visiting or being visited by people who don't have a clear wifi void).
