While petting and nibbling on some piggies I noticed that I couldn't rename some of them, namely those with names. But on the other hand I could rename named chickens. The only exceptions to this would be the piggies in my house.
There is a certain "cool-down" time on renaming (at least in public spaces) .. that is if the pig (or animal?) has been named or re-named within a certain time-frame, you or anyone else cannot change it. As I understand it, if an animal has not been named, you will get the "name" option.
I am not certain as to how long the "cool-down" period is, though.