I would find it very practically, if there were a possibility to rename the file that already exist in an folder on overwrite.
This is useful for copying newer versions of files somewhere and concurrently rename the already existing file in something like _old.
Maybe an also automatic way would be good, like: "Rename all already existing files . into _old."
That's not a bad idea. Being able to specify the pattern would be cool since you could then use the {date} and {time} codes.
It wouldn't be too hard to make some buttons which make doing this easier (but not automated) already. There's already a command to select the files in the destination which already exist in the source (based on names) so you could click that and then click a rename button to do it in a couple of steps. This wouldn't work very well when copying subdirectories, though. (Flat-mode helps a bit but if there's more than one file with the same name in the hierarchy then the results of the "select source in dest" command aren't always what you want.)
Having it as a feature of the file-replace dialog would be much, much nicer. Like the existing auto-rename option in the drop-down, except it renames the old file instead of the new.
(My idea of being able to specify a pattern could also apply to the existing auto-rename functionality as well as to your idea. I guess the auto-rename would use the current (1) etc. system unless the rename button's command had arguments to tell it otherwise.)