I don't believe there is a way within the application interface. There may be a way to script it either in InDesign or Acrobat.
If you don't get another useful reply, ask in the InDesign Scripting forum (for InDesign) or Creating, Editing, Exporting PDF forum (for Acrobat).
The file name is already included as part of the Page Info mark, but you can always add a slug area and add any bit of text you want. There's a file name variable you can use to automate, too.
Peter's suggestion is the way to go as far as I can tell. Only problem is that the text variable needs to be updated to the name of the PDF before you export.
In order not to have to enter the name twice I wrote a little script, that you can find here: Update Textvariable before Export | InDesign FAQ