I don't think is possible to edit the name of the backup (or even the swap) file. Because latter vim needs to know each backup belongs to each file.
But you can configure a backup directory. You will have one fixed directory adding the following lines to your .vimrc:
" Store swap files in fixed location, not current directory.<br/> set dir=~/.vimswap//,/var/tmp//,/tmp//,.
There's also methods to create a backup directory relative to file path. It's a little more complicated, and you can find how to in the following source:
http://vim.wikia.com/wiki/Remove_swap_and_backup_files_from_your_working_directory