Site name cannot be changed for different languages and domains Drupal 7
It seems that the option in Domain Access "Override site name with name of domain." was causing all the problems. It was so simple, but was driving me crazy. It was checked and thus if I tried to change the value elsewhere (in the Variables section or in the Site info section), nothing happened, the value was not changed. Unchecking this option ("Override site name...") in the Domain Access configuration, this variable "site_name" can have multiple values depending on language and domain combinations.