One thing I have noticed for SharePoint online is that it does not have an Advanced Search Web Part, well, at least by default.
I have been looking all around and even added a Search Center thinking it might be the reason my site does not have an Advanced Search Web Part but to no avail.
I then found an interesting page under my Master Page gallery, AdvancedSearchLayout.aspx.
When I opened the page I saw that it has an Advanced Search WebPart. I tried editing the page to see if my list got updated but unfortunately it still doesn't have an Advanced Search Web Part.
What I did was to export the web part by clicking the Web Part properties and click Export. It should download a .DWP file.
After downloading the file you can go back to your site, Edit a page, and click upload web part. Point to the .DWP file you downloaded earlier. After uploading the Advanced Search Web Part should now appear under the Imported Web