In a script, is there a way to get the name of a saved search ?
Not as far as I know.
You might consider other options for "saving" a search criteria where you save the criteria into a record used to repeat the find in the future. If you had the user enter/select criteria in a set of global fields while in Browse mode, a script could set up and perform the find. You can then copy the data from the global fields into such a record so that the criteria can be used again in the future.
For more on such scripted finds: See the first part of: Scripted Find Examples
Retrieving data ...