How to Edit List Views if Controls are Grey

For Custom Profile: 

1. From Setup, enter Profiles in the Quick Find box, then select Profiles.
2. Select the Profile of affected user.
3. Click on System Permissions.
4. Click on "Edit".
5. Check "Manage Public List Views" and "Create and Customize List Views".
6. Save.

For Standard Profile create Permission Sets: 

1. From Setup, enter Permission Sets in the Quick Find box, then select Permission Sets.
2. Click on New.
3. Enter your permission set information.
4. Select the types of users for the permission set.
When you create a permission set, you select a specific user or permission set license. If only users with one type of license can use the permission set, select the license that’s associated with the users. For example, to create a permission set for users with:

• the Salesforce license, select Salesforce. You can enable permissions only allowed in the Salesforce license.
• the Identity Connect permission set license, select Identity Connect. You can enable permissions only allowed in the Identity Connect license.
• select None as the License if you want to assign it to users with different license types.

5. Click System Permissions.
6. Click on "Edit".
7. Check "Manage Public List Views" and "Create and Customize List Views"
8. Save. 
9. Assign to target users.

The following video demonstrates how to create a Permission Set in Salesforce Lightning Experience, assign the required List View permissions, and assign it to a user.

Note: The end user cannot edit the Standard Recently Viewed list view for any tab or object. Administrator can configure it by going into the Object Manager for that object. Go to the search layouts section, change the Default Layout and change which columns appear in what order in the recently viewed tab.

See Also: 

Create Permission Sets

Create, Edit, Delete, and Assign a Permission Set