In order to Edit an Internal Custom Tab, a user must have the Store Tab permission with a Level of at least EDIT.
Steps to edit a Role and add the Permission:
1. Navigate to Setup > Users/Roles > Manage Roles.
2. Click Edit for the custom role.
3. Permissions tab > Lists subtab.
4. Add Store Tabs permission.
5. Set Level = Edit or Full.
6. Click Save.
Note: Edit level does not allow the user to delete the existing Tabs.