Using the administrator role, ensure that both permissions, Perform Search and Publish Search are set to Full permission level for the custom role.
1. Once set, have the custom role user log out and then log back in.
2. Then, the user may navigate again to Fixed Assets -> Lists -> Assets and click on Customize View.
3. Click More Options button which will provide more options to the user in customizing and saving their search.
4. Once done with customizing the search, click the Preview button.
5. After reviewing the results, click the Return To Criteria button found at the bottom of the results page. The user will then have the Save/Save & Run options for the search.