How to change the value of site_name to "My site name" in multilingual sites using Drush?
I understand your question as trying to use Drush for setting the site name, using Drush command like so:

drush variable-set site_name "Some name here"


And this for a multilingual site.

The equivalent of doing this via Drupal admin screens (instead of Drush), would require to first declare your site name as a "multilingual varialable". For more details on these kind of variables, and how to get this to work, refer to my answer at How to make content translation available for front page?.

In this case however, in "step 4" the variable to be selected is "Site name" (not "Default front frontpage"), within "Site information". While you're at it, you probably want to double check other similar variables like Site Slogan, etc.

If you try to do this using Drush, it won't work however, as explained in the issue about set i18n variables with drush? However, if you have the Variable module enabled, then according to Comment #3 in that issue, it should be possible to use Drush to achieve the equivalent result via Drush command like so (to set the Spanish language (es) to 'Spanish site'):

variable-realm-set language es site_name "Spanish Site"


Curious to some hear if in this case this Drush command resolved the issue here.