There are two possibilities:
(1) If the iPod was previously set to automatically sync to her iTunes library (versus the Manually manage music setting), you cannot add new content to the iPod without losing its current content. Automatic syncing means it is associated with only ONE iTunes library.
If this is the case, what you can do is offload the iPod's content to the computer's hard drive first, add the songs to your iTunes library, then sync the consolidated iTunes library to the iPod. You cannot transfer song files from iPod to computer using iTunes, However, there are third-party methods and utilities that can transfer from iPod to computer. If you do a Google search on something like "ipod transfer," you should get some links.
(2) If it is currently (already) using the manual setting, you should be able to connect the iPod to your iTunes library, and manually add more content. However, you cannot set it to automatically sync to your library, because that would remove content from the previous iTunes library. So, if you want to make it automatically sync to your iTunes library, you'll need to do what I described in (1) to offload the iPod's content.