Tamed mobs should have an option for moving around independently from the player.

Maybe villagers should have pets too?

Pets should have a home block. When the owner or a friend of the pet first enters a certain radius (of the pet), the pet “greets” them, following them around for a bit, then goes back to wandering around within the confines of their home block radius. Probably the “home” radius would be on a larger scale than the “detect player” radius - would have to play with it.

Different values could also be randomly generated for “home” and “detect player” radiuses do that not every pet seems as friendly as every other.