Use usermod to change the username and home directory.
Change user peter to paul:
usermod -l paul peter
grep paul /etc/passwd /etc/shadow # check
change home directory and move everything there (created if necessary)
usermod -m -d /home/paul paul
grep paul /etc/passwd # check
Change the group name, but don't use usermod as it requires you to first make another group (eg. GID 1001) and although it will update the GID for everything within the $HOME directory, you will have to search for and change the files and directories elsewhere with the old group ownership.
Instead, just change the group name manually.
You will have to change the filenames used by cron, at, and other services that use the username instead of the UID.
find /var -name "*peter*"
to find these files.
You will also have to change the config files used by some services such as you mail handling applications.
Finally, scripts using "/home/peter/" will need to be edited. Change occurences of "/home/peter/" to "$HOME".
It isn't a lot of work to change user identities, you just need to be thorough.