We've added a Search Box web part on a page. After editing the page, when a user tries to edit the Search Box web part, it gives him error "Sorry, Something went wrong". He has 'Design' permission on the site and page. This error happens only for this particular user but for his colleague who has same permission as this user, is able to edit the search box.
Go to the masterpage gallery and check the permissions for the library. The permissions shouldn't be inherited and the user should have "read"-permission. As default the "all authentificated user"-Group have read-permission to this library. I got the same error as you and it worked fine for me.