Do you mean you're extracting some PDF files from an archive (e.g. a zip file)? (If that, how are you doing the extraction? i.e. What are you clicking on or doing. That will affect what needs to be edited to make it behave differently.)
Or are you extracting the individual pages from a multi-page PDF file into separate PDF files? (If that, which tool/method are you using?)
[ul][li]Go to Settings -> File Types[/li]
[li]Expand the System File Types section and double-click the PDF file type[/li]
[li]Selecting the Events tab, and double-click the Drag-and-Drop event[/li]
[li]Set it to run Copy WHENEXISTS=rename[/li][/ul]
This has some drawbacks in that it will do the same for all PDF files that you drag to Opus. They will always be copied (not moved; unless you add the MOVE or MOVEWHENSAME arguments, at least), and they will always be auto-renamed if they clash.
I also have not been able to test that this will work with this particular case, because Adobe Reader does not let me drag individual pages out of documents and I do not have Acrobat Pro to test against. But, from your video, it looks like Acrobat is creating the PDF files somewhere and then dropping them on Opus the same as a file dragged & dropped from another Opus window or, say, Explorer. (Which is also why changing what happens when pages are dropped from Acrobat probably also has to change what happens when PDF files are dropped from anything else.)
Ideally, Acrobat itself would allow you to tell it to name the files it produces more sensibly, but I don't know if it does. I found this thread, which talks about a configuration page that may be worth looking at, and also discusses scripting (although it didn't seem to result in any scripts): answers.acrobatusers.com/How-I-e ... 33279.aspx
Another way you could handle it is to drop the files normally, without changing the file type's drag-and-drop event, and then create a button or hotkey in Opus which you can use to quickly give a better name to the files you just dropped. e.g. It could insert the date and time into the names (a command to do that is already in the menu which is joined to the Rename button on the default toolbars). You could then drop more files in without their names conflicting (since the old files have been renamed to something unique), and then hit the button/hotkey again to rename the new files, ready for more.
One last thing to note is that you can get quick access to the Rename New option in the Replace File dialog by holding down combinations of the Ctrl and Alt keys. That can mean you only have to hold a key down and click a button in the dialog, rather than open the menu. It saves a bit of time when doing it a lot, and is the easiest (though maybe not the best) of the possibilities to use.