Check out the Debian Live manual, especially the Using the space left on a USB stick and the Persistence sections, which explain creating a new partition and using it for persistence. Above links for the stable version in English, other combinations are also available from the project page (now redirect to the new homepage).
Main points:
* a special boot parameter to be specified at boot time: `persistence`
* the volume label for overlays must be `persistence` but it will be ignored unless it contains in its root a file named `persistence.conf`
* to make `/home` persistent, `persistence.conf` could simply contain:
/home
* neither `/lib`, `/lib/live` (or any of their sub-directories) nor `/` can be made persistent using custom mounts. As a workaround for this limitation you can add `/ union` to your persistence.conf file to achieve full persistence.