How to get the old villager back (I want them to remember me)? – Animal Crossing New Leaf

According to a wiki

It is obtained from visitors in the Happy Home Showcase you have streetpassed 9 or 10 times.

It can also be obtained one from Isabelle at the New Year's Eve countdown if the player speaks to her twice.

This FAQ on Reddit specifically around New Leaf says:

After you’ve had 4 villagers move out, the 16 most recent villagers to have left your town have the chance to show up on main street to do a little shopping every day. They just miss you! If you don’t want them there, get 16 more villagers to move out and the one you don’t like will be gone. This is also a good indicator of whether an old villager can move back into your town or not. The moment they stop appearing on main street, they’ll have cleared the 16 villager cycle and will be able to move back in to your town.

Now about moving in, I do know that you cannot convince then to move back even if you see them shopping about. You'll have to get them by random chance again or find the desired villager in a friend's town who is moving out.