I'm not sure the method you are using to load the iPod, manual or using automatic syncing. First, you may want to do a Restore on the iPod, in case there is data corruption on the iPod's storage, and that's why you have problems finding the song on the iPod through the iPod's display. The Restore button is on the iPod's Summary screen in iTunes. This will erase the iPod, reinstall its software, and set it to default settings. It will then have a "fresh start," and you can try syncing it as before.
If you have been using the manual method and want to use automatic syncing (which is often more convenient), here are some tips. First, if the iTunes sidebar (along left side of window) is hidden, from the menu bar, under View, select Show Sidebar. The iPod appears under DEVICES in the sidebar, when connected.
Select the iPod in the sidebar. The iPod's Summary screen appears to the right. Toward the top, there is a bar of buttons, starting with Summary; click on Music next to Summary. This Music screen is where you tell iTunes how to sync songs to your iPod.
Check the box for Sync Music. This setting enables automatic syncing. Unless you want the entire iTunes music library on the iPod, choose the option to sync Selected playlists, artists, albums, and genres. Then, select (checkmark) what you want on the iPod from the lists below. One way to set this up for your situation is to create one or more playlists for your daughter in your iTunes library, and put your daughter's songs on those playlists. Back on the iPod's Music screen, select those playlists under Playlists.
When you click Apply, the songs in your selection(s) sync to the iPod. Going forward, to update songs on the iPod, update those playlists (add/remove songs) in your iTunes library. The iPod does not need to be connected, so your daughter can be using it. The next time you connect the iPod (or click Sync if already connected), iTunes updates the iPod with the same changes automatically. And the songs may be easier to find (and listen to), if they are on playlists.
