New Ahrefs v3 API: We have upgraded to the new Ahrefs API on 18th October. All customers will need to re-authenticate the Ahrefs integration after the upgrade. Here is a blog post on the migration
To modify a search, you first need to select it from the "Saved Searches" drop down list at the top of the page. Once selected the Modify and Delete links will be shown next to the drop down.