Unfortunately Nautilus doesn't have that option.

Option 1: A different file manager

You could try another file manager like Dolphin.

(requires the Universe repository)

Option 2: Command-line

You can also use the command line program cp(1) with the backup option:

cp --backup -t DESTINATION SOURCE [SOURCE...]

This has the following effects which can be controlled with other options as described in the manual page of cp(1):

--backup[=CONTROL] ― make a backup of each existing destination file

-b ― like --backup but does not accept an argument

-S, --suffix=SUFFIX ― override the usual backup suffix

The backup suffix is ~, unless set with --suffix or SIMPLE_BACKUP_SUFFIX. The version control method may be selected via the --backup option or through the VERSION_CONTROL environment variable. Here are the values:

• none, off: never make backups (even if --backup is given)
• numbered, t: make numbered backups
• existing, nil: numbered if numbered backups exist, simple otherwise
• simple, never: always make simple backups

Example

cp --backup=existing --suffix=.orig -t ~/Videos ~/Music/*

This will copy all files in ~/Music to ~/Videos. If a file of the same name exists at the destination, it is renamed by appending .orig to its name as a backup. If a file with the same name as the backup exists, the backup is instead renamed by appending .1 and if that exists as well .2 and so forth. Only then is the source file copied to the destination.

If you want to copy files in subdirectories recursively use:

cp -R --backup=existing --suffix=.orig -t ~/Videos ~/Music

If you want a find solution, you could use this:

find parent -name "source" -type d -exec bash -c 'cd "$1"; mv * ..' bash {} \;

Explanation:

• find parent -name "source" -type d - For each directory named source in parent...
• -exec bash -c '...' bash {} \; - Call Bash with bash as $0 and the directory path as $1
• cd "$1"; mv * .. - cd into the directory; move all its contents up one level.
  • Alternative: mv "$1"/* "$1/.."

This is more or less based on dessert's answer.

Including hidden files

find parent -name "source" -type d -exec bash -c 'shopt -s dotglob; cd "$1"; mv * ..' bash {} \;

• shopt -s dotglob - Make globs include hidden files