Your mom will need to go into her Amazon account, and under Manage my Kindle, deregister it.
Then you go to your Amazon Account and register her kindle under your account.
Books on your Mom's account, already on the Kindle will remain there, but when/if they are deleted, they will be removed and no longer available. Any books you have in your account, will be available on both Kindles. Any books you have on your account, that is not on your Mom's Kindle, will show up under archive and will sync to the Kindle if you select the book title under Archive.
You will want to keep the Whispher Net (radio
) off, if one syncs their Kindle and is reading the same book ahead of the other, the book marks will advance to the "fastest" reader. Also, if one deletes a book on one Kindle, it will be deleted on the other when a sync is done - although the books will remain in your account under archive.