Wherever desired add in the constant STORE_NAME, though not sure what the correct variable name is if running a multi-store from a single database.
Code:
$file_name = $_SERVER['HTTP_HOST'].'_'.STORE_NAME.'_'.date("d-m-Y_H:i");
For example, but would also be a good idea to do some string formatting on it to remove spaces, and do something with apostrophes and quotes, etc... Not sure off top of my head, which function(s) would be best to use at this point, but I'm sure ZC has some built in to handle filenames.