The item require to put her to rest is a book that she read every night before going to bed. Here's the text for that book:
"This worn, sturdy tome is no spellbook, as you might expect a Kirin Tor mage to carry. The pages of the volume are covered with hand-copied stories, accompanied by colorful illustrations.
The tome seems to be far older than its former owner and may have even been a family heirloom. A powerful tie to the book may have been the last thread tying Belmara's spirit to the ruins of Kirin'Var. Returning it to its proper place on the bookshelf in her quarters might help appease her restless spirit"
Putting her soul to rest will make the spirit neutral towards you.