I happened upon the phrasal verb in the movie Back To the Future. Here is the context:
Biff: Do you realize what would happen if I hand in my reports in your handwriting. I'll get fired. You wouldn't want that to happen would you? Would you?
George: Of course not, Biff, now I wouldn't want that to happen. Now, uh, I'll finish those reports up tonight, and I'll run them on over first thing tomorrow, alright?
I can understand that run on over means to deliver, what I cannot get is what the on adds to the meaning.