In step 3 the "LABEL=..." entry in fstab makes it so that whatever has the label MY_BACKUPS will be seen as the proper device regardless of whether it is sdc, sdc1, sdd, etc....?

0. Buy a USB2 disk that has enough space to hold all the stuff in your home
dir plus at least a few G more (data tends to get bigger with time, of
course)

1. Plug this disk in.  Usually, removable disks have 1 partition of type FAT32
or NTFS covering their whole space.  (Check that this is the case, if not,
something weird may be going on.)

2. Make a filesystem with a label on this partition.  "mke2fs -j -L MY_BACKUPS
/dev/sdN1" .  Find what N is by looking at the output of dmesg | tail.

3. Make an entry for the partition you made in your /etc/fstab :

LABEL=MY_BACKUPS  /mnt/backup  ext3  noauto,users,noatime  0  0

4. As root, mkdir /mnt/backup if it doesn't exist, then mount this partition
on /mnt/backup , mkdir /mnt/backup/USER , and chown USER /mnt/backup/USER .

5. Make a shell script sort of like this:

#!/bin/bash
if [[ $1 == '--help' || $1 == '-h' ]] ; then
    echo "backs up ~USER to backup drive."
    exit;
fi

if mount | grep /mnt/backup > /dev/null ; then
    rsync -av --delete-after /home/USER/ /mnt/backup/USER
else
    echo "backup disk not mounted.  Trying to mount it."
    mount /mnt/backup
    if mount | grep /mnt/backup > /dev/null ; then
         echo "Is the disk plugged in?  Can't mount, bailing."
         exit 1
    fi
    rsync -av --delete-after /home/USER/ /mnt/backup/USER
    umount /mnt/backup
fi

6. Any time you want to make a backup, plug your disk in, and run that shell
script.  The initial rsync will take some time.  Subsequent rsyncs will take a
couple of minutes.