There are two different customizers in this game, a character and a creature, and using either of them will unlock this achievement.
The creature customizer located at Beorn’s Homestead along the western edge of Mirkwood. Fast travel to that location, and when facing the eagle statue, turn to the right and into the building with the three vertical windows on the left (these glow orange at night). The entrance is on the right of this building.
To activate the customizer, walk onto the floor plate of the desired creature (horse, pony, warg or troll [which requires a big-fig character to unlock first]). Select your creature to customize, and once the doors open, change anything about the creature (name, color, etc.). The achievement will unlock soon after.
While at Beorn’s Homestead, if you have 45,000 studs, you can unlock “Beorn Again.”
The character customizer is in Lake-town. From the eagle statue, walk right from the statue and the shop is the first building on the right.