"It's either all your library, or all your mum's library. You can't just add songs from another library."
Not true.
The first time you connect an iPod to iTunes it comes up with an option to allow you to manually manage your music or automatically sync it. Most people will pick the automatic option and I think that's even the default. That for most people makes life very easy, it will make your iPod look like the iTunes you connect it to automatically. So you add a tune to your iTunes and then it's on your iPod, you delete it from iTunes and it's gone on your iPod. Very simple.
However that's not what you want.
So when asked that question either say you don't want to automatically sync your music or you want to manually manage it (I can't quite recall which way round the question is asked).
That will stop iTunes making the connected iPod look like your iTunes. You can then pick whatever music you do want form your library and drag and drop it onto the iPod yourself.
However the next time your Mum connects the iPod to her computer with it's automatic options set iTunes will see that the iPod has music on it that it doesn't have in it's libary and it will do exactly what it's been told to do and make the iPod look like iTunes, which will mean the music gets deleted.
So you'll have to set the iPod up on your Mums PC to be manually managed to (connect the iPod, click on it in iTunes and check the "manually manage music" box) before you copy your music on to it yourself.
So you've got two options....
Option A
- Connect your Mum's iPod to your Mum's PC and put it into manual mode.
- Connect your Mum's iPod to your PC, tell it to manually manage music when it asks and then drag the music on to it.
- From now on your Mum will have to manually manage music which means when she wants to add stuff she'll have to drag it onto her iPod.
Option B
- Copy the music from your computer into your Mum's iTunes. Use a memory stick for example.
- Connect your Mum's iPod and it will all work out.
So you can mix libraries with iPods and you can use them on multiple computers. But you have to manually manage your music, which means a bit more work for you.
Automatically syncing your music makes looking after an iPod a lot easier but stops you from being able to do more complex things such as having multiple computers each with different things on them.