I don't know of a way to have R give you the script name. But you could include it at the top of the script - like so:
Script.Name<-c("current name")
Then using the Sys.time() and paste() commands you could make the object you want:
Final.Name<-paste(Script.Name, Sys.time(), sep="_")
If you just want the date of the run, use Sys.Date() instead of Sys.time().