In my advanced search properties, I can't find my site column that I have crawled and mapped to a managed property.
Looking up the list of managed properties requires more rights to the Search Service Application than most users have. So in order to avoid that problem MS has decided not to try to do that in the advanced search box web part.
Instead the list of properties are controlled by configuration of the web part.
On order for your managed property to show up you have to go to the advanced search page, put it into edit mode, select "Edit web part" on the "Advanced Search Box". Below the "Properties" header you then have the "Properties" property.
Click the elipsis (...) and copy the content of the textbox into a proper editor.
Copy the xml back, OK, OK, Save, Test and Publish